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Chapter 9: Q. 11 (page 227)

Evaluate the dot product A鈫�路B鈫�if

a. A鈫�=4i^-2j^and B鈫�=-2i^-3j^.

b. A鈫�=-4i^+2j^andB鈫�=2i^+4j^.

Short Answer

Expert verified

a). A鈫�路B鈫�=-2

b).A鈫�路B鈫�=0

Step by step solution

01

Step 1. Given Information

We have A鈫�=4i^-2j^and B鈫�=-2i^-3j^

alsoA鈫�=-4i^+2j^andB鈫�=2i^+4j^

02

Step 2. Part a). Dot product A→·B→ if A→=4i^-2j^ and B→=-2i^-3j^

The dot product for these two vectors is given A鈫�路B鈫�=AxBx+AyByin this form

Now, we put the x-components and y-components of the two vectors

A鈫�路B鈫�=AxBx+AyBy=(4i^)(-2i^)+(-2j^)(-3j^)=-8i^2+6j^2

The magnitudes of i^2=1and j^2=1. So, the dot product will be

A鈫�路B鈫�=-8i^2+6j^2=-8+6=-2

03

Step 3. Part b). Dot product A→·B→ if A→=-4i^+2j^and B→=2i^+4j^

We use the same step as in part (a) to get the dot product by putting the x-components and y-components of the two vector

A鈫�路B鈫�=AxBx+AyBy=(-4i^)(2i^)+(2j^)(4j^)=-8i^2+8j^2=-8+8=0

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