Concept explainers
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. If r′(t) = 0, then r(t) = 〈a, b, c〉, where a, b, and c are real numbers.
- e. The parameterized curve r(t) = 〈5 cos t, 12 cos t, 13 sin t〉 has arc length as a parameter.
- f. The position vector and the principal unit normal are always parallel on a smooth curve.
a.
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.
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.
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 vector u and v are parallel.
Therefore, the given statement is true.
d.
Answer to Problem 1RE
The given statement is true.
Explanation of Solution
Given:
“If
Calculation:
Consider
Thus, the vector
Therefore, the given statement is true.
e.
Answer to Problem 1RE
The given statement is false.
Explanation of Solution
Given:
“The parameterized curve
Formula used:
Suppose
Calculation:
Differentiate
Compute
Since
Therefore, the given statement is false.
f.
Answer to Problem 1RE
The given statement is false.
Explanation of Solution
Given:
“The position vector and the principal unit normal are always parallel on a smooth curve.”
Formula used:
Suppose r is a smooth parameterized curve and s is the arc length.
The unit tangent vector T is
The principal unit normal vector is
Calculation:
Counter example
Consider
Differentiate
Use magnitude formula to obtain the value of
On further simplification,
Use unit tangent formula to compute
Thus, the unit tangent vector
Differentiate
Use magnitude formula to obtain the value of
On further simplification,
Use principal unit normal formula to compute the value of
Thus, the principal unit normal vector
It is observed that the position vector and the principal unit normal vector are not equal.
Therefore, the given statement is false.
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Chapter 11 Solutions
Calculus: Early Transcendentals (2nd Edition)
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