   Chapter 12.3, Problem 43E

Chapter
Section
Textbook Problem

# Find the scalar and vector projections of b onto a.43. a = 3i −3j + k, b = 2i + 4j − k

To determine

To find: The scalar projection of b onto a and the vector projection of b onto a.

Explanation

Given:

a=3i3j+k (1)

b=2i+4jk (2)

Formula:

Consider three dimensional vectors a and b as follows.

a=a1i+a2j+a3k (3)

b=b1i+b2j+b3k (4)

Write the expression for scalar projection of b onto a.

compab=ab|a| (5)

Consider a general expression to find the dot product between two three-dimensional vectors.

ab=a1,a2,a3b1,b2,b3

ab=a1b1+a2b2+a3b3 (6)

Consider a general expression to find the magnitude of a three dimensional vector that is a=a1,a2,a3 .

|a|=a12+a22+a32 (7)

Write the expression for vector projection of b onto a.

projab=ab|a|2a (8)

Compare equation (1) and (3).

a=3,3,1

Compare equation (1) and (3).

b=2,4,1

In equation (6), substitute 3 for a1 , –3 for a2 , 1 for a3 , 2 for b1 , 4 for b2 and –1 for b3

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