Chapter 9: Q. 12 (page 228)

Evaluate the dot product $\vec{A} \cdot \vec{B}$ if
a). role="math" localid="1647759176820" $\vec{A} = 3 \hat{i} + 4 \hat{j}$ and $\vec{B} = 2 \hat{i} - 6 \hat{j}$
b). $\vec{A} = 3 \hat{i} - 2 \hat{j}$ and $\vec{B} = 6 \hat{i} + 4 \hat{j}$

Short Answer

Expert verified

a). $\vec{A} \cdot \vec{B} = - 18$

b). $\vec{A} \cdot \vec{B} = 10$

Step by step solution

Step 1. Given information

We have,

$\vec{A} = 3 \hat{i} + 4 \hat{j}$ and $\vec{B} = 2 \hat{i} - 6 \hat{j}$

also $\vec{A} = 3 \hat{i} - 2 \hat{j}$ and $\vec{B} = 6 \hat{i} + 4 \hat{j}$

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

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

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

$\vec{A} \cdot \vec{B} = A_{x} B_{x} + A_{y} B_{y} = (3 \hat{i}) (2 \hat{i}) + (4 \hat{j}) (- 6 \hat{j}) = 6 {\hat{i}}^{2} - 24 {\hat{j}}^{2}$

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

$\vec{A} \cdot \vec{B} = 6 {\hat{i}}^{2} - 24 {\hat{j}}^{2} = 6 - 24 = - 18$

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

$\vec{A} \cdot \vec{B} = A_{x} B_{x} + A_{y} B_{y} = (3 \hat{i}) (6 \hat{i}) + (- 2 \hat{j}) (4 \hat{j}) = 18 {\hat{i}}^{2} - 8 {\hat{j}}^{2} = 18 - 8 = 10$

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Evaluate the dot product $\vec{A} \cdot \vec{B}$ if
a). role="math" localid="1647759176820" $\vec{A} = 3 \hat{i} + 4 \hat{j}$ and $\vec{B} = 2 \hat{i} - 6 \hat{j}$
b). $\vec{A} = 3 \hat{i} - 2 \hat{j}$ and $\vec{B} = 6 \hat{i} + 4 \hat{j}$

Short Answer

Step by step solution

Step 1. Given information

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

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

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Evaluate the dot product A→·B→ifa). role="math" localid="1647759176820" A→=3i^+4j^and B→=2i^-6j^b).A→=3i^-2j^andB→=6i^+4j^

Short Answer

Step by step solution

Step 1. Given information

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

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

One App. One Place for Learning.

Most popular questions from this chapter

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Torque and Rotational Motion

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Study anywhere. Anytime. Across all devices.

Evaluate the dot product $\vec{A} \cdot \vec{B}$ if
a). role="math" localid="1647759176820" $\vec{A} = 3 \hat{i} + 4 \hat{j}$ and $\vec{B} = 2 \hat{i} - 6 \hat{j}$
b). $\vec{A} = 3 \hat{i} - 2 \hat{j}$ and $\vec{B} = 6 \hat{i} + 4 \hat{j}$