**Title: Vector Analysis and Operations****Consider Two Vectors:**- Vector $ \vec{A} $: Magnitude = 16.5 meters, Angle = 48.0°- Vector $ \vec{B} $: Components = $(-12.7 \, \hat{i} + 2.30 \, \hat{j})$ meters**Determine the x and y Components of $ \vec{A} $:**- $ A_x = $ _____ meters- $ A_y = $ _____ meters**Determine the x and y Components of $ \vec{B} $:**- $ B_x = $ _____ meters- $ B_y = $ _____ meters**Calculate Resultant Vector $ \vec{H} $:**Let $ \vec{H} = \vec{B} - 3 \vec{A} $. Determine $ \vec{H} $ and express it in unit vector form:- $ \vec{H} = ($ _____ $ \hat{i} +$ _____ $ \hat{j})$ meters**Determine the Magnitude and Direction of $ \vec{H} $:**- Magnitude $ |\vec{H}| = $ _____ meters - Direction = _____ °**Instructions:**- Use trigonometric functions to resolve vector $ \vec{A} $ into its x and y components.- Perform vector subtraction and scalar multiplication as necessary for $ \vec{H} $.- Calculate the magnitude of $ \vec{H} $ using the Pythagorean theorem.- Find the direction of $ \vec{H} $ by computing the angle with respect to the positive x-axis.

The given vectors, A→=16.5 m at 48°B→=-12.7 i^+2.30 j^ m The x and y components of A vector,…

Answered: Consider two vectors, A = 16.5 m at…

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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Question

**Title: Vector Analysis and Operations**

**Consider Two Vectors:**
- Vector $ \vec{A} $: Magnitude = 16.5 meters, Angle = 48.0°
- Vector $ \vec{B} $: Components = $(-12.7 \, \hat{i} + 2.30 \, \hat{j})$ meters

**Determine the x and y Components of $ \vec{A} $:**
- $ A_x = $ _____ meters
- $ A_y = $ _____ meters

**Determine the x and y Components of $ \vec{B} $:**
- $ B_x = $ _____ meters
- $ B_y = $ _____ meters

**Calculate Resultant Vector $ \vec{H} $:**
Let $ \vec{H} = \vec{B} - 3 \vec{A} $. Determine $ \vec{H} $ and express it in unit vector form:

- $ \vec{H} = ($ _____ $ \hat{i} +$ _____ $ \hat{j})$ meters

**Determine the Magnitude and Direction of $ \vec{H} $:**
- Magnitude $ |\vec{H}| = $ _____ meters
- Direction = _____ °

**Instructions:**
- Use trigonometric functions to resolve vector $ \vec{A} $ into its x and y components.
- Perform vector subtraction and scalar multiplication as necessary for $ \vec{H} $.
- Calculate the magnitude of $ \vec{H} $ using the Pythagorean theorem.
- Find the direction of $ \vec{H} $ by computing the angle with respect to the positive x-axis.

Expert Solution

Step 1

The given vectors,

$\begin{array}{rcl} \vec{A} & = & 16.5 m at 48 ° \\ \vec{B} & = & (- 12.7 \hat{i} + 2.30 \hat{j}) m \end{array}$

The x and y components of A vector,

$\begin{array}{rcl} A_{x} & = & (16.5 m) \cos (48 °) \\ = & 11.04 m \\ A_{y} & = & (16.5 m) \sin (48 °) \\ = & 12.26 m \end{array}$

Step 2

So, the vector A in unit vector form

$\vec{A} = (11.04 \hat{i} + 12.26 \vec{j}) m$

The x and y components of B vector,

$\begin{array}{rcl} B_{x} & = & - 12.7 m \\ B_{y} & = & 2.30 m \end{array}$

Step 3

The H vector,

$\vec{H} = \vec{B} - 3 \vec{A}$

Substitute,

$\begin{array}{rcl} \vec{H} & = & (- 12.7 \hat{i} + 2.30 \vec{j}) m - 3 (11.04 \hat{i} + 12.26 \vec{j}) m \\ = & (- 45.82 \hat{i} - 34.48 \vec{j}) m \end{array}$

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