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Chapter 14: Problem 3

On a diagram show the arbitrary vectors \(p\) and q. Then show the following: (a) \(p+q\) (d) \(4 q\) (e) \(-2 q\) (c) \(q-p\).

Short Answer

Expert verified
Question: Represent the following vector operations for arbitrary vectors p and q: (a) p + q, (b) 4q, (c) -2q, and (d) q - p. Answer: (a) To represent p + q, place the initial point of vector q at the end point of vector p and draw a vector from the initial point of p to the end point of q. (b) To represent 4q, multiply the magnitude of q by 4 while keeping the direction unchanged. (c) To represent -2q, multiply the magnitude of q by 2 but reverse its direction. (d) To represent q - p, first find -p by rotating vector p by 180 degrees, then place the initial point of vector -p at the end point of vector q, and draw a vector from the initial point of q to the end point of -p.

Step by step solution

01

Vector Addition (p + q)

To find the p + q, place the initial point of vector q at the end point of vector p. Now, draw a vector from the initial point of vector p to the end point of vector q. This new vector represents the sum p + q.
02

Multiplication by Scalar (4q)

To multiply vector q by a scalar, we must change the magnitude of the vector q while keeping its direction unchanged. In this case, the scalar is 4, so the new vector (4q) will have 4 times the length of the original vector q in the same direction as q.
03

Multiplication by Scalar (-2q)

To multiply vector q by a scalar, we must change the magnitude of the vector q while keeping its direction unchanged. In this case, the scalar is -2, so the new vector (-2q) will have 2 times the length of the original vector q but in the opposite direction as q because of the negative sign.
04

Vector Subtraction (q - p)

To subtract p from q, we start by finding -p. To find -p, rotate vector p by 180 degrees. Now, we can treat q - p as an addition of vectors q and -p. Hence, place the initial point of vector -p at the end point of vector q, and then draw a vector from the initial point of vector q to the end point of vector -p. This new vector represents the difference q - p.

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Most popular questions from this chapter

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