Chapter 13, Problem 1RE

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. Given two vectors u and v, it is always true that 2u + v = v + 2u.

b. The vector in the direction of u with the length of v equals the vector in the direction of v with the length of u.

c. If u ≠ 0 and u + v = 0, then u and v are parallel.

d. The lines x = 3 + t, y = 4 + 2t, z = 2 −t and x = 2t, y = 4t, z = t are parallel.

e. The lines x = 3 + t, y = 4 + 2t, z = 2 − t and the plane x + 2y + 5z = 3 are parallel.

f. There is always a plane orthogonal to both of two distinct intersecting planes.

To determine

Whether the given statement is true or not and give an explanation or counterexample.

Answer to Problem 1RE

The given statement is true.

Explanation of Solution

Given:

“Given two vectors u and v, it is always true that $2u+v=v+2u$”.

Formula used:

Suppose the vectors $u=\langle a,b\rangle \text{ and }v=\langle c,d\rangle$ and the scalar is c. Then,

Vector addition is $u+v=\langle a+c,b+d\rangle$.

Scalar multiplication is $cu=\langle ca,cb\rangle$.

Commutative property $u+v=v+u$.

Calculation:

Suppose $u=\langle a,b\rangle \text{ and }v=\langle c,d\rangle$.

Use vector addition and scalar multiplication to compute the value of $2u+v$.

$\begin{array}{c}2u+v=2\langle a,b\rangle +\langle c,d\rangle \\ =\langle 2a,2b\rangle +\langle c,d\rangle \\ =\langle 2a+c,2b+d\rangle \end{array}$

Thus, the component of the vector, $2u+v$ is $\langle 2a+c,2b+d\rangle$.                               (1)

Use vector addition and scalar multiplication to compute the value of $v+2u$.

$\begin{array}{c}v+2u=\langle c,d\rangle +2\langle a,b\rangle \\ =\langle c,d\rangle +\langle 2a,2b\rangle \\ =\langle c+2a,d+2b\rangle \\ =\langle 2a+c,2b+d\rangle \left[\because \text{vector addition is commutative}\right]\end{array}$

Thus, the component of the vector, $v+2u$ is $\langle 2a+c,2b+d\rangle$.                               (2)

From the equations (1) and (2), it is observed that $2u+v=v+2u$.

Therefore, the given statement is true.

To determine

Whether the given statement is true or not and give an explanation or counterexample.

Answer to Problem 1RE

The given statement is false.

Explanation of Solution

Given:

“The vector in the direction of u with the length of v equals the vector in the direction of v with the length of u”.

Formula used:

Suppose the two vectors are u and v.

The unit vector in the direction of u with the length of v is $\left|v\right|\frac{u}{\left|u\right|}$.

Calculation:

Suppose $u=\langle a,b\rangle \text{ and }v=\langle c,d\rangle$.

Let x be the unit vector in the direction of u with the length of v.

Use the above mentioned formula to compute the vector x.

$\begin{array}{c}x=\left|\langle c,d\rangle \right|\frac{\langle a,b\rangle }{\left|\langle a,b\rangle \right|}\\ =\sqrt{c^2+d^2}\frac{\langle a,b\rangle }{\sqrt{a^2+b^2}}\end{array}$

Thus, the vector x is $\sqrt{c^2+d^2}\frac{\langle a,b\rangle }{\sqrt{a^2+b^2}}$.                                                                (1)

Let y be the unit vector in the direction of v with the length of u.

Use the above mentioned formula to compute the vector y.

$\begin{array}{c}y=\left|\langle a,b\rangle \right|\frac{\langle c,d\rangle }{\left|\langle c,d\rangle \right|}\\ =\sqrt{a^2+b^2}\frac{\langle c,d\rangle }{\sqrt{c^2+d^2}}\end{array}$

Thus, the vector y is $\sqrt{a^2+b^2}\frac{\langle c,d\rangle }{\sqrt{c^2+d^2}}$.                                                                (2)

From the equations (1) and (2), it is observed that both the vectors are not equal.

Therefore, the given statement is false.

To determine

Whether the given statement is true or not and give an explanation or counterexample.

Answer to Problem 1RE

The given statement is true.

Explanation of Solution

Given:

“If $u\ne 0$ and $u+v=0$, then u and v are parallel”.

Result used:

The vectors u and v are said to be parallel vectors, if one vector is the scalar multiple of the other vector.

Calculation:

Consider $u+v=0$ and simplify.

$\begin{array}{l}u+v=0\\ u=-v\end{array}$

This implies that the vector u is −1 times the vector v. By the result of parallel vectors, the two vectors u and v are parallel.

Therefore, the given statement is true.

To determine

Whether the given statement is true or not and give an explanation or counterexample.

Answer to Problem 1RE

The given statement is false.

Explanation of Solution

Given:

“The lines $x=3+t,y=4+2t,z=2-t$ and $x=2t,y=4t,z=t$ are parallel.”

Calculation:

Consider the parametric equation of the lines $x=3+t,y=4+2t,z=2-t$ and $x=2t,y=4t,z=t$.

Note that, the direction vector of a parametric line is a coefficient of t in the x, y and z direction.

The direction vectors of the above line equations are ${\text{v}}_{\text{1}}=\langle 1,2,-1\rangle$ and ${\text{v}}_{\text{2}}=\langle 2,4,1\rangle$.

Since the vectors ${\text{v}}_{\text{1}}$ and ${\text{v}}_{\text{2}}$ are not a constant multiple of each other, the vectors are not parallel.

Therefore, the given statement is false.

To determine

Whether the given statement is true or not and give an explanation or counterexample.

Answer to Problem 1RE

The given statement is true.

Explanation of Solution

Given:

“The line $x=3+t,y=4+2t,z=2-t$ and the plane $x+2y+5z=3$. are parallel.”

Result used:

“If the line and plane are parallel then the dot product of the direction vector of the line and normal to the plane is zero”.

Calculation:

Consider the line and plane equation $x=3+t,y=4+2t,z=2-t$ and $x+2y+5z=3$.

Note that, the direction vector of a line is the coefficient of t in the x, y and z direction.

The direction vector of the above line equation $x=3+t,y=4+2t,z=2-t$ is ${\text{v}}_{\text{1}}=\langle 1,2,-1\rangle$.

The normal vector of the plane equation $x+2y+5z=3$ is ${\text{v}}_{\text{2}}=\langle 1,2,5\rangle$.

Obtain the dot product of the vectors ${\text{v}}_{\text{1}}=\langle 1,2,-1\rangle$ and ${\text{v}}_{\text{2}}=\langle 1,2,5\rangle$.

$\begin{array}{c}{\text{v}}_{\text{1}}\cdot {\text{v}}_{\text{2}}=\langle 1,2,-1\rangle \cdot \langle 1,2,5\rangle \\ =1+4-5\\ =0\end{array}$

By the Result, it can be conclude that, the given line and plane are parallel.

Therefore, the given statement is true.

To determine

Whether the given statement is true or not and give an explanation or counterexample.

Answer to Problem 1RE

The given statement is true.

Explanation of Solution

Given:

“There is always a plane orthogonal to both of two distinct intersecting planes.”

Definition used:

Cross product:

“Given two nonzero vectors u and v in R³, the cross product $\text{u}\times \text{v}=\left|\text{u}\right|\left|\text{v}\right|\sin \theta$, where $0\le \theta \le \pi$ is the angle between u and v. The direction of the vector $\text{u}\times \text{v}$ is orthogonal to both the vector u and v”.

Interpretation:

Assume that, P₁ and P₂ are two planes and they intersect at the lines L.

Consider the normal vectors v₁ and v₂ of the planes P₁ and P₂.

By the above definition, the direction of ${\text{v}}_{\text{1}}\times {\text{v}}_{\text{2}}$ is orthogonal to both the vectors v₁ and v₂.

That is, the vector ${\text{v}}_{\text{1}}\times {\text{v}}_{\text{2}}$ is normal to both the planes P₁ and P₂.

It is always easy to find a plane P₃ which has the normal vector ${\text{v}}_{\text{1}}\times {\text{v}}_{\text{2}}$.

The plane P₃ will be orthogonal to both P₁ and P₂.

Therefore, the given statement is true.