Concept explainers
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. Given two
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.
a.
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
Formula used:
Suppose the vectors
Vector addition is
Scalar multiplication is
Commutative property
Calculation:
Suppose
Use vector addition and scalar multiplication to compute the value of
Thus, the component of the vector,
Use vector addition and scalar multiplication to compute the value of
Thus, the component of the vector,
From the equations (1) and (2), it is observed that
Therefore, the given statement is true.
b.
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
Calculation:
Suppose
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.
Thus, the vector x is
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.
Thus, the vector y is
From the equations (1) and (2), it is observed that both the vectors are not equal.
Therefore, the given statement is false.
c.
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
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
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.
d.
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
Calculation:
Consider the parametric equation of the lines
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
Since the vectors
Therefore, the given statement is false.
e.
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
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
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
The normal vector of the plane equation
Obtain the dot product of the vectors
By the Result, it can be conclude that, the given line and plane are parallel.
Therefore, the given statement is true.
f.
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
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
That is, the vector
It is always easy to find a plane P3 which has the normal vector
The plane P3 will be orthogonal to both P1 and P2.
Therefore, the given statement is true.
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