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Chapter 9: Q. 12 (page 228)

Evaluate the dot product A鈫�路B鈫�if

a). role="math" localid="1647759176820" A鈫�=3i^+4j^and B鈫�=2i^-6j^

b).A鈫�=3i^-2j^andB鈫�=6i^+4j^

Short Answer

Expert verified

a). A鈫�路B鈫�=-18

b).A鈫�路B鈫�=10

Step by step solution

01

Step 1. Given information

We have,

A鈫�=3i^+4j^andB鈫�=2i^-6j^

also A鈫�=3i^-2j^andB鈫�=6i^+4j^

02

Step 2. Part a). dot product A→·B→ if A→=3i^+4j^ and B→=2i^-6j^

The dot product for these two vectors is in the form

A鈫�路B鈫�=AxBx+AyBy

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

A鈫�路B鈫�=AxBx+AyBy=(3i^)(2i^)+(4j^)(-6j^)=6i^2-24j^2

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

A鈫�路B鈫�=6i^2-24j^2=6-24=-18

03

Step 3. Part b). Dot product A→·B→ if A→=3i^-2j^ and B→=6i^+4j^

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

A鈫�路B鈫�=AxBx+AyBy=(3i^)(6i^)+(-2j^)(4j^)=18i^2-8j^2=18-8=10

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