Mark each of the following statements true or false: For vectors u , v , and w in ℝ n , if u + w = v + w then u = v . For vectors u , v , and w in ℝ n , if u ⋅ w = v ⋅ w , then u = v For vectors u , v , and w in ℝ 3 , if u is orthogonal to v , and v is orthogonal to w , then u is orthogonal to w . In ℝ 3 , if a line l is parallel to a plane P , then a di- rection vector d for l is parallel to a normal vector n for P . In ℝ 3 , if a line l is perpendicular to a plane P , then a direction vector d for l is a parallel to a normal vector n for P . ℝ 3 , if two planes are not parallel, then they must intersect in a line . In ℝ 3 , if two lines are not parallel, then they must intersect in a point . If v is a binary vector such that v ⋅ v = 0 , then v = 0 . In ℤ 5 , if a b = 0 then either a = 0 or b = 0 . ln ℤ 6 , if a b = 0 then either a = 0 or b = 0

Question

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Answer 1

Textbook Question

Chapter 1, Problem 1RQ

Mark each of the following statements true or false:

For vectors u , v , and w in ℝ n , if u + w = v + w then u = v .
For vectors u , v , and w in ℝ n , if u ⋅ w = v ⋅ w , then u = v
For vectors u , v , and w in ℝ 3 , if u is orthogonal to v , and v is orthogonal to w , then u is orthogonal to w .
In ℝ 3 , if a line l is parallel to a plane P , then a di- rection vector d for l is parallel to a normal vector n for P .
In ℝ 3 , if a line l is perpendicular to a plane P , then a direction vector d for l is a parallel to a normal vector n for P .
ℝ 3 , if two planes are not parallel, then they must intersect in a line .
In ℝ 3 , if two lines are not parallel, then they must intersect in a point .
If v is a binary vector such that v ⋅ v = 0 , then v = 0 .
In ℤ 5 , if a b = 0 then either a = 0 or b = 0 .
ln ℤ 6 , if a b = 0 then either a = 0 or b = 0

(a)

Expert Solution

To determine

Whether the statement, “if u+w=v+w, then u=v” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

Given that, u+w=v+w

adding (-w) to both the sides, we get

u+w+-w=v+w+(-w)

or, u+w-w=v+w-w using associative property

or, u+0=v+0

or, u=v

Thus, u=v.

Therefore, the given statement is true.

(b)

Expert Solution

To determine

Whether the statement, “if u∙w=v∙w, then u=v” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the vectors u=1,1,1, v=1,0,1 and w=(1,0,0).

u∙w=1,1,1∙1,0,0=11+10+10=1

v∙w=1,0,1∙1,0,0=11+00+10=1

Thus, u∙w=v∙w but u≠v.

Therefore, the given statement is false.

(c)

Expert Solution

To determine

Whether the statement, “if u is orthogonal to v, and v is orthogonal to w, then u is

orthogonal to w ” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the vectors u=1,0,0, v=0,0,1 and w=(2,0,0).

u∙v=1,0,0∙0,0,1=10+00+01=0

as u∙v=0, thus u is orthogonal to v.

v∙w=0,0,1∙2,0,0=02+00+10=0

as v∙w=0, thus v is orthogonal to w.

u∙w=1,0,0∙2,0,0=12+00+00=2

as u∙v=2≠0, thus u is not orthogonal to w.

Therefore, the given statement is false.

(d)

Expert Solution

To determine

Whether the statement, “if a line l is parallel to plane P, then a direction vector d for l is

parallel to normal vector n for P ” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below statement disproves the given statement.

if a line l is parallel to plane P, then a direction vector d for l is orthogonal to

normal vector n for P

Mathematically, d∙n=0

Therefore, the given statement is false.

(e)

Expert Solution

To determine

Whether the statement, “if a line l is perpendicular to plane P, then a direction vector d

for l is parallel to normal vector n for P ” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

As the line line l is perpendicular to plane P, therefore the direction vector of

the line and normal vector of the plane must be parallel to each other.

Therefore, the given statement is true.

(f)

Expert Solution

To determine

Whether the statement, “if two planes are not parallel, then they must intersect

in a line” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

Two planes always intersect in a line, as long as they are not parallel.

Therefore, the given statement is true.

(g)

Expert Solution

To determine

Whether the statement, “if two lines are not parallel, then they must intersect

in a point” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Skew lines are non-parallel lines that do not intersect each other.

For example, consider the Z-axis and the line x+y=2, z=0.

Both the lines are not parallel to each other, and they do not intersect at any point.

Therefore, the given statement is false.

(h)

Expert Solution

To determine

Whether the statement, “if v is a binary vector such that v∙v=0, then v=0” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the binary vector v=1,1,0

We know that, for binary vectors 1+1=0

Thus, v∙v=1,1,0∙1,1,0=11+11+00=1+1=0

But, v≠0

Therefore, the given statement is false.

(i)

Expert Solution

To determine

Whether the statement, “In Z5, if ab=0 then either a=0 or b=0.” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

We can prove the above statement by proving that Z5 does not have zero divisors.

Here, Z5=0,1,2,3,4.

Now, 0∙1=0, 0∙2=0, 0∙3=0, 0∙4=0, 1∙2=2, 1∙3=3, 1∙4=4,

2∙3=1, 2∙4=3, 3∙4=2.

Thus, there does not exist zero divisors.

Hence, in Z5, if ab=0 then either a=0 or b=0.

Therefore, the given statement is true.

(j)

Expert Solution

To determine

Whether the statement, “In Z6, if ab=0 then either a=0 or b=0.” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

Counter Example:

Let a=3, b=2∈Z6.

Now, ab=6=0 in Z6, but neither a≠0 nor b≠0.

Therefore, the given statement is false.

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Answer 2

Textbook Question

Chapter 1, Problem 1RQ

Mark each of the following statements true or false:

For vectors u , v , and w in ℝ n , if u + w = v + w then u = v .
For vectors u , v , and w in ℝ n , if u ⋅ w = v ⋅ w , then u = v
For vectors u , v , and w in ℝ 3 , if u is orthogonal to v , and v is orthogonal to w , then u is orthogonal to w .
In ℝ 3 , if a line l is parallel to a plane P , then a di- rection vector d for l is parallel to a normal vector n for P .
In ℝ 3 , if a line l is perpendicular to a plane P , then a direction vector d for l is a parallel to a normal vector n for P .
ℝ 3 , if two planes are not parallel, then they must intersect in a line .
In ℝ 3 , if two lines are not parallel, then they must intersect in a point .
If v is a binary vector such that v ⋅ v = 0 , then v = 0 .
In ℤ 5 , if a b = 0 then either a = 0 or b = 0 .
ln ℤ 6 , if a b = 0 then either a = 0 or b = 0

(a)

Expert Solution

To determine

Whether the statement, “if u+w=v+w, then u=v” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

Given that, u+w=v+w

adding (-w) to both the sides, we get

u+w+-w=v+w+(-w)

or, u+w-w=v+w-w using associative property

or, u+0=v+0

or, u=v

Thus, u=v.

Therefore, the given statement is true.

(b)

Expert Solution

To determine

Whether the statement, “if u∙w=v∙w, then u=v” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the vectors u=1,1,1, v=1,0,1 and w=(1,0,0).

u∙w=1,1,1∙1,0,0=11+10+10=1

v∙w=1,0,1∙1,0,0=11+00+10=1

Thus, u∙w=v∙w but u≠v.

Therefore, the given statement is false.

(c)

Expert Solution

To determine

Whether the statement, “if u is orthogonal to v, and v is orthogonal to w, then u is

orthogonal to w ” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the vectors u=1,0,0, v=0,0,1 and w=(2,0,0).

u∙v=1,0,0∙0,0,1=10+00+01=0

as u∙v=0, thus u is orthogonal to v.

v∙w=0,0,1∙2,0,0=02+00+10=0

as v∙w=0, thus v is orthogonal to w.

u∙w=1,0,0∙2,0,0=12+00+00=2

as u∙v=2≠0, thus u is not orthogonal to w.

Therefore, the given statement is false.

(d)

Expert Solution

To determine

Whether the statement, “if a line l is parallel to plane P, then a direction vector d for l is

parallel to normal vector n for P ” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below statement disproves the given statement.

if a line l is parallel to plane P, then a direction vector d for l is orthogonal to

normal vector n for P

Mathematically, d∙n=0

Therefore, the given statement is false.

(e)

Expert Solution

To determine

Whether the statement, “if a line l is perpendicular to plane P, then a direction vector d

for l is parallel to normal vector n for P ” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

As the line line l is perpendicular to plane P, therefore the direction vector of

the line and normal vector of the plane must be parallel to each other.

Therefore, the given statement is true.

(f)

Expert Solution

To determine

Whether the statement, “if two planes are not parallel, then they must intersect

in a line” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

Two planes always intersect in a line, as long as they are not parallel.

Therefore, the given statement is true.

(g)

Expert Solution

To determine

Whether the statement, “if two lines are not parallel, then they must intersect

in a point” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Skew lines are non-parallel lines that do not intersect each other.

For example, consider the Z-axis and the line x+y=2, z=0.

Both the lines are not parallel to each other, and they do not intersect at any point.

Therefore, the given statement is false.

(h)

Expert Solution

To determine

Whether the statement, “if v is a binary vector such that v∙v=0, then v=0” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the binary vector v=1,1,0

We know that, for binary vectors 1+1=0

Thus, v∙v=1,1,0∙1,1,0=11+11+00=1+1=0

But, v≠0

Therefore, the given statement is false.

(i)

Expert Solution

To determine

Whether the statement, “In Z5, if ab=0 then either a=0 or b=0.” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

We can prove the above statement by proving that Z5 does not have zero divisors.

Here, Z5=0,1,2,3,4.

Now, 0∙1=0, 0∙2=0, 0∙3=0, 0∙4=0, 1∙2=2, 1∙3=3, 1∙4=4,

2∙3=1, 2∙4=3, 3∙4=2.

Thus, there does not exist zero divisors.

Hence, in Z5, if ab=0 then either a=0 or b=0.

Therefore, the given statement is true.

(j)

Expert Solution

To determine

Whether the statement, “In Z6, if ab=0 then either a=0 or b=0.” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

Counter Example:

Let a=3, b=2∈Z6.

Now, ab=6=0 in Z6, but neither a≠0 nor b≠0.

Therefore, the given statement is false.

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Answer 3

Textbook Question

Chapter 1, Problem 1RQ

Mark each of the following statements true or false:

For vectors u , v , and w in ℝ n , if u + w = v + w then u = v .
For vectors u , v , and w in ℝ n , if u ⋅ w = v ⋅ w , then u = v
For vectors u , v , and w in ℝ 3 , if u is orthogonal to v , and v is orthogonal to w , then u is orthogonal to w .
In ℝ 3 , if a line l is parallel to a plane P , then a di- rection vector d for l is parallel to a normal vector n for P .
In ℝ 3 , if a line l is perpendicular to a plane P , then a direction vector d for l is a parallel to a normal vector n for P .
ℝ 3 , if two planes are not parallel, then they must intersect in a line .
In ℝ 3 , if two lines are not parallel, then they must intersect in a point .
If v is a binary vector such that v ⋅ v = 0 , then v = 0 .
In ℤ 5 , if a b = 0 then either a = 0 or b = 0 .
ln ℤ 6 , if a b = 0 then either a = 0 or b = 0

(a)

Expert Solution

To determine

Whether the statement, “if u+w=v+w, then u=v” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

Given that, u+w=v+w

adding (-w) to both the sides, we get

u+w+-w=v+w+(-w)

or, u+w-w=v+w-w using associative property

or, u+0=v+0

or, u=v

Thus, u=v.

Therefore, the given statement is true.

(b)

Expert Solution

To determine

Whether the statement, “if u∙w=v∙w, then u=v” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the vectors u=1,1,1, v=1,0,1 and w=(1,0,0).

u∙w=1,1,1∙1,0,0=11+10+10=1

v∙w=1,0,1∙1,0,0=11+00+10=1

Thus, u∙w=v∙w but u≠v.

Therefore, the given statement is false.

(c)

Expert Solution

To determine

Whether the statement, “if u is orthogonal to v, and v is orthogonal to w, then u is

orthogonal to w ” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the vectors u=1,0,0, v=0,0,1 and w=(2,0,0).

u∙v=1,0,0∙0,0,1=10+00+01=0

as u∙v=0, thus u is orthogonal to v.

v∙w=0,0,1∙2,0,0=02+00+10=0

as v∙w=0, thus v is orthogonal to w.

u∙w=1,0,0∙2,0,0=12+00+00=2

as u∙v=2≠0, thus u is not orthogonal to w.

Therefore, the given statement is false.

(d)

Expert Solution

To determine

Whether the statement, “if a line l is parallel to plane P, then a direction vector d for l is

parallel to normal vector n for P ” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below statement disproves the given statement.

if a line l is parallel to plane P, then a direction vector d for l is orthogonal to

normal vector n for P

Mathematically, d∙n=0

Therefore, the given statement is false.

(e)

Expert Solution

To determine

Whether the statement, “if a line l is perpendicular to plane P, then a direction vector d

for l is parallel to normal vector n for P ” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

As the line line l is perpendicular to plane P, therefore the direction vector of

the line and normal vector of the plane must be parallel to each other.

Therefore, the given statement is true.

(f)

Expert Solution

To determine

Whether the statement, “if two planes are not parallel, then they must intersect

in a line” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

Two planes always intersect in a line, as long as they are not parallel.

Therefore, the given statement is true.

(g)

Expert Solution

To determine

Whether the statement, “if two lines are not parallel, then they must intersect

in a point” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Skew lines are non-parallel lines that do not intersect each other.

For example, consider the Z-axis and the line x+y=2, z=0.

Both the lines are not parallel to each other, and they do not intersect at any point.

Therefore, the given statement is false.

(h)

Expert Solution

To determine

Whether the statement, “if v is a binary vector such that v∙v=0, then v=0” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the binary vector v=1,1,0

We know that, for binary vectors 1+1=0

Thus, v∙v=1,1,0∙1,1,0=11+11+00=1+1=0

But, v≠0

Therefore, the given statement is false.

(i)

Expert Solution

To determine

Whether the statement, “In Z5, if ab=0 then either a=0 or b=0.” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

We can prove the above statement by proving that Z5 does not have zero divisors.

Here, Z5=0,1,2,3,4.

Now, 0∙1=0, 0∙2=0, 0∙3=0, 0∙4=0, 1∙2=2, 1∙3=3, 1∙4=4,

2∙3=1, 2∙4=3, 3∙4=2.

Thus, there does not exist zero divisors.

Hence, in Z5, if ab=0 then either a=0 or b=0.

Therefore, the given statement is true.

(j)

Expert Solution

To determine

Whether the statement, “In Z6, if ab=0 then either a=0 or b=0.” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

Counter Example:

Let a=3, b=2∈Z6.

Now, ab=6=0 in Z6, but neither a≠0 nor b≠0.

Therefore, the given statement is false.

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Answer 4

Textbook Question

Chapter 1, Problem 1RQ

Mark each of the following statements true or false:

For vectors u , v , and w in ℝ n , if u + w = v + w then u = v .
For vectors u , v , and w in ℝ n , if u ⋅ w = v ⋅ w , then u = v
For vectors u , v , and w in ℝ 3 , if u is orthogonal to v , and v is orthogonal to w , then u is orthogonal to w .
In ℝ 3 , if a line l is parallel to a plane P , then a di- rection vector d for l is parallel to a normal vector n for P .
In ℝ 3 , if a line l is perpendicular to a plane P , then a direction vector d for l is a parallel to a normal vector n for P .
ℝ 3 , if two planes are not parallel, then they must intersect in a line .
In ℝ 3 , if two lines are not parallel, then they must intersect in a point .
If v is a binary vector such that v ⋅ v = 0 , then v = 0 .
In ℤ 5 , if a b = 0 then either a = 0 or b = 0 .
ln ℤ 6 , if a b = 0 then either a = 0 or b = 0

(a)

Expert Solution

To determine

Whether the statement, “if u+w=v+w, then u=v” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

Given that, u+w=v+w

adding (-w) to both the sides, we get

u+w+-w=v+w+(-w)

or, u+w-w=v+w-w using associative property

or, u+0=v+0

or, u=v

Thus, u=v.

Therefore, the given statement is true.

(b)

Expert Solution

To determine

Whether the statement, “if u∙w=v∙w, then u=v” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the vectors u=1,1,1, v=1,0,1 and w=(1,0,0).

u∙w=1,1,1∙1,0,0=11+10+10=1

v∙w=1,0,1∙1,0,0=11+00+10=1

Thus, u∙w=v∙w but u≠v.

Therefore, the given statement is false.

(c)

Expert Solution

To determine

Whether the statement, “if u is orthogonal to v, and v is orthogonal to w, then u is

orthogonal to w ” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the vectors u=1,0,0, v=0,0,1 and w=(2,0,0).

u∙v=1,0,0∙0,0,1=10+00+01=0

as u∙v=0, thus u is orthogonal to v.

v∙w=0,0,1∙2,0,0=02+00+10=0

as v∙w=0, thus v is orthogonal to w.

u∙w=1,0,0∙2,0,0=12+00+00=2

as u∙v=2≠0, thus u is not orthogonal to w.

Therefore, the given statement is false.

(d)

Expert Solution

To determine

Whether the statement, “if a line l is parallel to plane P, then a direction vector d for l is

parallel to normal vector n for P ” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below statement disproves the given statement.

if a line l is parallel to plane P, then a direction vector d for l is orthogonal to

normal vector n for P

Mathematically, d∙n=0

Therefore, the given statement is false.

(e)

Expert Solution

To determine

Whether the statement, “if a line l is perpendicular to plane P, then a direction vector d

for l is parallel to normal vector n for P ” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

As the line line l is perpendicular to plane P, therefore the direction vector of

the line and normal vector of the plane must be parallel to each other.

Therefore, the given statement is true.

(f)

Expert Solution

To determine

Whether the statement, “if two planes are not parallel, then they must intersect

in a line” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

Two planes always intersect in a line, as long as they are not parallel.

Therefore, the given statement is true.

(g)

Expert Solution

To determine

Whether the statement, “if two lines are not parallel, then they must intersect

in a point” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Skew lines are non-parallel lines that do not intersect each other.

For example, consider the Z-axis and the line x+y=2, z=0.

Both the lines are not parallel to each other, and they do not intersect at any point.

Therefore, the given statement is false.

(h)

Expert Solution

To determine

Whether the statement, “if v is a binary vector such that v∙v=0, then v=0” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the binary vector v=1,1,0

We know that, for binary vectors 1+1=0

Thus, v∙v=1,1,0∙1,1,0=11+11+00=1+1=0

But, v≠0

Therefore, the given statement is false.

(i)

Expert Solution

To determine

Whether the statement, “In Z5, if ab=0 then either a=0 or b=0.” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

We can prove the above statement by proving that Z5 does not have zero divisors.

Here, Z5=0,1,2,3,4.

Now, 0∙1=0, 0∙2=0, 0∙3=0, 0∙4=0, 1∙2=2, 1∙3=3, 1∙4=4,

2∙3=1, 2∙4=3, 3∙4=2.

Thus, there does not exist zero divisors.

Hence, in Z5, if ab=0 then either a=0 or b=0.

Therefore, the given statement is true.

(j)

Expert Solution

To determine

Whether the statement, “In Z6, if ab=0 then either a=0 or b=0.” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

Counter Example:

Let a=3, b=2∈Z6.

Now, ab=6=0 in Z6, but neither a≠0 nor b≠0.

Therefore, the given statement is false.

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Answer 5

Textbook Question

Answer 6

Textbook Question

Answer 7

(a)

Expert Solution

To determine

Whether the statement, “if u+w=v+w, then u=v” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

Given that, u+w=v+w

adding (-w) to both the sides, we get

u+w+-w=v+w+(-w)

or, u+w-w=v+w-w using associative property

or, u+0=v+0

or, u=v

Thus, u=v.

Therefore, the given statement is true.

(b)

Expert Solution

To determine

Whether the statement, “if u∙w=v∙w, then u=v” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the vectors u=1,1,1, v=1,0,1 and w=(1,0,0).

u∙w=1,1,1∙1,0,0=11+10+10=1

v∙w=1,0,1∙1,0,0=11+00+10=1

Thus, u∙w=v∙w but u≠v.

Therefore, the given statement is false.

(c)

Expert Solution

To determine

Whether the statement, “if u is orthogonal to v, and v is orthogonal to w, then u is

orthogonal to w ” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the vectors u=1,0,0, v=0,0,1 and w=(2,0,0).

u∙v=1,0,0∙0,0,1=10+00+01=0

as u∙v=0, thus u is orthogonal to v.

v∙w=0,0,1∙2,0,0=02+00+10=0

as v∙w=0, thus v is orthogonal to w.

u∙w=1,0,0∙2,0,0=12+00+00=2

as u∙v=2≠0, thus u is not orthogonal to w.

Therefore, the given statement is false.

(d)

Expert Solution

To determine

Whether the statement, “if a line l is parallel to plane P, then a direction vector d for l is

parallel to normal vector n for P ” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below statement disproves the given statement.

if a line l is parallel to plane P, then a direction vector d for l is orthogonal to

normal vector n for P

Mathematically, d∙n=0

Therefore, the given statement is false.

(e)

Expert Solution

To determine

Whether the statement, “if a line l is perpendicular to plane P, then a direction vector d

for l is parallel to normal vector n for P ” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

As the line line l is perpendicular to plane P, therefore the direction vector of

the line and normal vector of the plane must be parallel to each other.

Therefore, the given statement is true.

(f)

Expert Solution

To determine

Whether the statement, “if two planes are not parallel, then they must intersect

in a line” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

Two planes always intersect in a line, as long as they are not parallel.

Therefore, the given statement is true.

(g)

Expert Solution

To determine

Whether the statement, “if two lines are not parallel, then they must intersect

in a point” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Skew lines are non-parallel lines that do not intersect each other.

For example, consider the Z-axis and the line x+y=2, z=0.

Both the lines are not parallel to each other, and they do not intersect at any point.

Therefore, the given statement is false.

(h)

Expert Solution

To determine

Whether the statement, “if v is a binary vector such that v∙v=0, then v=0” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the binary vector v=1,1,0

We know that, for binary vectors 1+1=0

Thus, v∙v=1,1,0∙1,1,0=11+11+00=1+1=0

But, v≠0

Therefore, the given statement is false.

(i)

Expert Solution

To determine

Whether the statement, “In Z5, if ab=0 then either a=0 or b=0.” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

We can prove the above statement by proving that Z5 does not have zero divisors.

Here, Z5=0,1,2,3,4.

Now, 0∙1=0, 0∙2=0, 0∙3=0, 0∙4=0, 1∙2=2, 1∙3=3, 1∙4=4,

2∙3=1, 2∙4=3, 3∙4=2.

Thus, there does not exist zero divisors.

Hence, in Z5, if ab=0 then either a=0 or b=0.

Therefore, the given statement is true.

(j)

Expert Solution

To determine

Whether the statement, “In Z6, if ab=0 then either a=0 or b=0.” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

Counter Example:

Let a=3, b=2∈Z6.

Now, ab=6=0 in Z6, but neither a≠0 nor b≠0.

Therefore, the given statement is false.

Answer 8

(a)

Expert Solution

To determine

Whether the statement, “if u+w=v+w, then u=v” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

Given that, u+w=v+w

adding (-w) to both the sides, we get

u+w+-w=v+w+(-w)

or, u+w-w=v+w-w using associative property

or, u+0=v+0

or, u=v

Thus, u=v.

Therefore, the given statement is true.

Answer 9

(a)

Expert Solution

Answer 10

(a)

Expert Solution

Answer 11

Expert Solution

Answer 12

To determine

Whether the statement, “if u+w=v+w, then u=v” is true or false.

Answer 13

Answer to Problem 1RQ

The statement is true.

Answer 14

Explanation of Solution

Given that, u+w=v+w

adding (-w) to both the sides, we get

u+w+-w=v+w+(-w)

or, u+w-w=v+w-w using associative property

or, u+0=v+0

or, u=v

Thus, u=v.

Therefore, the given statement is true.

Answer 15

(b)

Expert Solution

To determine

Whether the statement, “if u∙w=v∙w, then u=v” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the vectors u=1,1,1, v=1,0,1 and w=(1,0,0).

u∙w=1,1,1∙1,0,0=11+10+10=1

v∙w=1,0,1∙1,0,0=11+00+10=1

Thus, u∙w=v∙w but u≠v.

Therefore, the given statement is false.

Answer 16

(b)

Expert Solution

Answer 17

(b)

Expert Solution

Answer 18

Expert Solution

Answer 19

To determine

Whether the statement, “if u∙w=v∙w, then u=v” is true or false.

Answer 20

Answer to Problem 1RQ

The statement is false.

Answer 21

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the vectors u=1,1,1, v=1,0,1 and w=(1,0,0).

u∙w=1,1,1∙1,0,0=11+10+10=1

v∙w=1,0,1∙1,0,0=11+00+10=1

Thus, u∙w=v∙w but u≠v.

Therefore, the given statement is false.

Answer 22

(c)

Expert Solution

To determine

Whether the statement, “if u is orthogonal to v, and v is orthogonal to w, then u is

orthogonal to w ” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the vectors u=1,0,0, v=0,0,1 and w=(2,0,0).

u∙v=1,0,0∙0,0,1=10+00+01=0

as u∙v=0, thus u is orthogonal to v.

v∙w=0,0,1∙2,0,0=02+00+10=0

as v∙w=0, thus v is orthogonal to w.

u∙w=1,0,0∙2,0,0=12+00+00=2

as u∙v=2≠0, thus u is not orthogonal to w.

Therefore, the given statement is false.

Answer 23

(c)

Expert Solution

Answer 24

(c)

Expert Solution

Answer 25

Expert Solution

Answer 26

To determine

Whether the statement, “if u is orthogonal to v, and v is orthogonal to w, then u is

orthogonal to w ” is true or false.

Answer 27

Answer to Problem 1RQ

The statement is false.

Answer 28

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the vectors u=1,0,0, v=0,0,1 and w=(2,0,0).

u∙v=1,0,0∙0,0,1=10+00+01=0

as u∙v=0, thus u is orthogonal to v.

v∙w=0,0,1∙2,0,0=02+00+10=0

as v∙w=0, thus v is orthogonal to w.

u∙w=1,0,0∙2,0,0=12+00+00=2

as u∙v=2≠0, thus u is not orthogonal to w.

Therefore, the given statement is false.

Answer 29

(d)

Expert Solution

To determine

Whether the statement, “if a line l is parallel to plane P, then a direction vector d for l is

parallel to normal vector n for P ” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below statement disproves the given statement.

if a line l is parallel to plane P, then a direction vector d for l is orthogonal to

normal vector n for P

Mathematically, d∙n=0

Therefore, the given statement is false.

Answer 30

(d)

Expert Solution

Answer 31

(d)

Expert Solution

Answer 32

Expert Solution

Answer 33

To determine

Whether the statement, “if a line l is parallel to plane P, then a direction vector d for l is

parallel to normal vector n for P ” is true or false.

Answer 34

Answer to Problem 1RQ

The statement is false.

Answer 35

Explanation of Solution

The given statement is false as the below statement disproves the given statement.

if a line l is parallel to plane P, then a direction vector d for l is orthogonal to

normal vector n for P

Mathematically, d∙n=0

Therefore, the given statement is false.

Answer 36

(e)

Expert Solution

To determine

Whether the statement, “if a line l is perpendicular to plane P, then a direction vector d

for l is parallel to normal vector n for P ” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

As the line line l is perpendicular to plane P, therefore the direction vector of

the line and normal vector of the plane must be parallel to each other.

Therefore, the given statement is true.

Answer 37

(e)

Expert Solution

Answer 38

(e)

Expert Solution

Answer 39

Expert Solution

Answer 40

To determine

Whether the statement, “if a line l is perpendicular to plane P, then a direction vector d

for l is parallel to normal vector n for P ” is true or false.

Answer 41

Answer to Problem 1RQ

The statement is true.

Answer 42

Explanation of Solution

As the line line l is perpendicular to plane P, therefore the direction vector of

the line and normal vector of the plane must be parallel to each other.

Therefore, the given statement is true.

Answer 43

(f)

Expert Solution

To determine

Whether the statement, “if two planes are not parallel, then they must intersect

in a line” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

Two planes always intersect in a line, as long as they are not parallel.

Therefore, the given statement is true.

Answer 44

(f)

Expert Solution

Answer 45

(f)

Expert Solution

Answer 46

Expert Solution

Answer 47

To determine

Whether the statement, “if two planes are not parallel, then they must intersect

in a line” is true or false.

Answer 48

Answer to Problem 1RQ

The statement is true.

Answer 49

Explanation of Solution

Two planes always intersect in a line, as long as they are not parallel.

Therefore, the given statement is true.

Answer 50

(g)

Expert Solution

To determine

Whether the statement, “if two lines are not parallel, then they must intersect

in a point” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Skew lines are non-parallel lines that do not intersect each other.

For example, consider the Z-axis and the line x+y=2, z=0.

Both the lines are not parallel to each other, and they do not intersect at any point.

Therefore, the given statement is false.

Answer 51

(g)

Expert Solution

Answer 52

(g)

Expert Solution

Answer 53

Expert Solution

Answer 54

To determine

Whether the statement, “if two lines are not parallel, then they must intersect

in a point” is true or false.

Answer 55

Answer to Problem 1RQ

The statement is false.

Answer 56

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Skew lines are non-parallel lines that do not intersect each other.

For example, consider the Z-axis and the line x+y=2, z=0.

Both the lines are not parallel to each other, and they do not intersect at any point.

Therefore, the given statement is false.

Answer 57

(h)

Expert Solution

To determine

Whether the statement, “if v is a binary vector such that v∙v=0, then v=0” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the binary vector v=1,1,0

We know that, for binary vectors 1+1=0

Thus, v∙v=1,1,0∙1,1,0=11+11+00=1+1=0

But, v≠0

Therefore, the given statement is false.

Answer 58

(h)

Expert Solution

Answer 59

(h)

Expert Solution

Answer 60

Expert Solution

Answer 61

To determine

Whether the statement, “if v is a binary vector such that v∙v=0, then v=0” is true or false.

Answer 62

Answer to Problem 1RQ

The statement is false.

Answer 63

Explanation of Solution

The given statement is false as the below example disproves the given statement.

Consider the binary vector v=1,1,0

We know that, for binary vectors 1+1=0

Thus, v∙v=1,1,0∙1,1,0=11+11+00=1+1=0

But, v≠0

Therefore, the given statement is false.

Answer 64

(i)

Expert Solution

To determine

Whether the statement, “In Z5, if ab=0 then either a=0 or b=0.” is true or false.

Answer to Problem 1RQ

The statement is true.

Explanation of Solution

We can prove the above statement by proving that Z5 does not have zero divisors.

Here, Z5=0,1,2,3,4.

Now, 0∙1=0, 0∙2=0, 0∙3=0, 0∙4=0, 1∙2=2, 1∙3=3, 1∙4=4,

2∙3=1, 2∙4=3, 3∙4=2.

Thus, there does not exist zero divisors.

Hence, in Z5, if ab=0 then either a=0 or b=0.

Therefore, the given statement is true.

Answer 65

(i)

Expert Solution

Answer 66

(i)

Expert Solution

Answer 67

Expert Solution

Answer 68

To determine

Whether the statement, “In Z5, if ab=0 then either a=0 or b=0.” is true or false.

Answer 69

Answer to Problem 1RQ

The statement is true.

Answer 70

Explanation of Solution

We can prove the above statement by proving that Z5 does not have zero divisors.

Here, Z5=0,1,2,3,4.

Now, 0∙1=0, 0∙2=0, 0∙3=0, 0∙4=0, 1∙2=2, 1∙3=3, 1∙4=4,

2∙3=1, 2∙4=3, 3∙4=2.

Thus, there does not exist zero divisors.

Hence, in Z5, if ab=0 then either a=0 or b=0.

Therefore, the given statement is true.

Answer 71

(j)

Expert Solution

To determine

Whether the statement, “In Z6, if ab=0 then either a=0 or b=0.” is true or false.

Answer to Problem 1RQ

The statement is false.

Explanation of Solution

Counter Example:

Let a=3, b=2∈Z6.

Now, ab=6=0 in Z6, but neither a≠0 nor b≠0.

Therefore, the given statement is false.

Answer 72

(j)

Expert Solution

Answer 73

(j)

Expert Solution

Answer 74

Expert Solution

Answer 75

To determine

Whether the statement, “In Z6, if ab=0 then either a=0 or b=0.” is true or false.

Answer 76

Answer to Problem 1RQ

The statement is false.

Answer 77

Explanation of Solution

Counter Example:

Let a=3, b=2∈Z6.

Now, ab=6=0 in Z6, but neither a≠0 nor b≠0.

Therefore, the given statement is false.

Answer 78

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Chapter 1 Solutions