Chapter 9: Q. 11 (page 227)

Evaluate the dot product $\vec{A} \cdot \vec{B}$ if
a. $\vec{A} = 4 \hat{i} - 2 \hat{j}$ and $\vec{B} = - 2 \hat{i} - 3 \hat{j}$ .
b. $\vec{A} = - 4 \hat{i} + 2 \hat{j}$ and $\vec{B} = 2 \hat{i} + 4 \hat{j}$ .

Short Answer

Expert verified

a). $\vec{A} \cdot \vec{B} = - 2$

b). $\vec{A} \cdot \vec{B} = 0$

Step by step solution

Step 1. Given Information

We have $\vec{A} = 4 \hat{i} - 2 \hat{j}$ and $\vec{B} = - 2 \hat{i} - 3 \hat{j}$

also $\vec{A} = - 4 \hat{i} + 2 \hat{j}$ and $\vec{B} = 2 \hat{i} + 4 \hat{j}$

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 $\vec{A} \cdot \vec{B} = A_{x} B_{x} + A_{y} B_{y}$ in this form

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

$\vec{A} \cdot \vec{B} = A_{x} B_{x} + A_{y} B_{y} = (4 \hat{i}) (- 2 \hat{i}) + (- 2 \hat{j}) (- 3 \hat{j}) = - 8 {\hat{i}}^{2} + 6 {\hat{j}}^{2}$

The magnitudes of ${\hat{i}}^{2} = 1$ and ${\hat{j}}^{2} = 1$ . So, the dot product will be

$\vec{A} \cdot \vec{B} = - 8 {\hat{i}}^{2} + 6 {\hat{j}}^{2} = - 8 + 6 = - 2$

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

$\vec{A} \cdot \vec{B} = A_{x} B_{x} + A_{y} B_{y} = (- 4 \hat{i}) (2 \hat{i}) + (2 \hat{j}) (4 \hat{j}) = - 8 {\hat{i}}^{2} + 8 {\hat{j}}^{2} = - 8 + 8 = 0$

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Evaluate the dot product $\vec{A} \cdot \vec{B}$ if
a. $\vec{A} = 4 \hat{i} - 2 \hat{j}$ and $\vec{B} = - 2 \hat{i} - 3 \hat{j}$ .
b. $\vec{A} = - 4 \hat{i} + 2 \hat{j}$ and $\vec{B} = 2 \hat{i} + 4 \hat{j}$ .

Short Answer

Step by step solution

Step 1. Given Information

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

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

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Evaluate the dot product A→·B→if a. A→=4i^-2j^and B→=-2i^-3j^. b. A→=-4i^+2j^andB→=2i^+4j^.

Short Answer

Step by step solution

Step 1. Given Information

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

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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Physics Textbooks

Nuclear Physics

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Study anywhere. Anytime. Across all devices.

Evaluate the dot product $\vec{A} \cdot \vec{B}$ if
a. $\vec{A} = 4 \hat{i} - 2 \hat{j}$ and $\vec{B} = - 2 \hat{i} - 3 \hat{j}$ .
b. $\vec{A} = - 4 \hat{i} + 2 \hat{j}$ and $\vec{B} = 2 \hat{i} + 4 \hat{j}$ .