### Vector Operations and Directions#### Given Vectors:You have two vectors:- Vector \(\vec{A}\) = \(3\hat{i} + 4\hat{j}\)- Vector \(\vec{B}\) = \(6\hat{i} - 5\hat{j}\)#### Problem Statement:"What is the direction of \(3\vec{A} + 2\vec{B}\)?"#### Solution:1. **Calculate \(3\vec{A}\):**\[3\vec{A} = 3(3\hat{i} + 4\hat{j}) = 9\hat{i} + 12\hat{j}\]2. **Calculate \(2\vec{B}\):**\[2\vec{B} = 2(6\hat{i} - 5\hat{j}) = 12\hat{i} - 10\hat{j}\]3. **Sum the scaled vectors:**\[3\vec{A} + 2\vec{B} = (9\hat{i} + 12\hat{j}) + (12\hat{i} - 10\hat{j}) = (9 + 12)\hat{i} + (12 - 10)\hat{j} = 21\hat{i} + 2\hat{j}\]4. **Calculate the direction:**The direction \(\theta\) of the vector \( \vec{C} = 21\hat{i} + 2\hat{j}\) is given by:\[\theta = \tan^{-1} \left( \frac{2}{21} \right)\]Use a calculator to find the arctangent:\[\theta \approx \tan^{-1}(0.0952) \approx 5.43^\circ\]Thus, the direction of \(3\vec{A} + 2\vec{B}\) is approximately \(5.43^\circ\) above the positive \(x\)-axis.

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You have two vectors. Vector A - 31 +43 and vector B = 62 - 53. What is the direction of 3A + 2B?

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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Scalars and Vectors

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Question

### Vector Operations and Directions

#### Given Vectors:
You have two vectors:
- Vector \(\vec{A}\) = \(3\hat{i} + 4\hat{j}\)
- Vector \(\vec{B}\) = \(6\hat{i} - 5\hat{j}\)

#### Problem Statement:
"What is the direction of \(3\vec{A} + 2\vec{B}\)?"

#### Solution:

1. **Calculate \(3\vec{A}\):**

\[
3\vec{A} = 3(3\hat{i} + 4\hat{j}) = 9\hat{i} + 12\hat{j}
\]

2. **Calculate \(2\vec{B}\):**

\[
2\vec{B} = 2(6\hat{i} - 5\hat{j}) = 12\hat{i} - 10\hat{j}
\]

3. **Sum the scaled vectors:**

\[
3\vec{A} + 2\vec{B} = (9\hat{i} + 12\hat{j}) + (12\hat{i} - 10\hat{j}) = (9 + 12)\hat{i} + (12 - 10)\hat{j} = 21\hat{i} + 2\hat{j}
\]

4. **Calculate the direction:**

The direction \(\theta\) of the vector \( \vec{C} = 21\hat{i} + 2\hat{j}\) is given by:

\[
\theta = \tan^{-1} \left( \frac{2}{21} \right)
\]

Use a calculator to find the arctangent:

\[
\theta \approx \tan^{-1}(0.0952) \approx 5.43^\circ
\]

Thus, the direction of \(3\vec{A} + 2\vec{B}\) is approximately \(5.43^\circ\) above the positive \(x\)-axis.

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