   Chapter 12.3, Problem 42E

Chapter
Section
Textbook Problem

# Find the scalar and vector projections of b onto a.42. a = ⟨−1, 4, 8⟩, b = ⟨12, 1, 2⟩

To determine

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

Explanation

Given:

a=1,4,8 and b=12,1,2

Formula:

Write the expression for scalar projection of b onto a.

compab=ab|a| (1)

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

ab=a1,a2,a3b1,b2,b3

ab=a1b1+a2b2+a3b3 (2)

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

|a|=a12+a22+a32 (3)

Write the expression for vector projection of b onto a.

projab=ab|a|2a (4)

In equation (2), substitute –1 for a1 , 4 for a2 ,8 for a3 , 12 for b1 ,1 for b2 and 2 for b3 .

ab=(1)(12)+(4)(1)+(8)(2)=12+4+16=8

In equation (3), substitute –1 for a1 , 4 for a2 and 8 for a3

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