Q: In the fourth image, if $$\|\vec{P}\| = 100$$, $$\|\vec{K}\| = 150$$, $$\|\vec{L}\| = 90$$, and $$\|\vec{N}\| = 120$$, which equation gives the $$x-$$component of $$\vec{R} = \vec{P} + \vec{K} + \vec{L} + \vec{N}$$?

$$R_x = P\cos\alpha + K\cos\beta + L\cos\theta + N\cos\mu$$

Q: A soccer player runs 12 m straight, then turns 30° to the left and runs 10 m. What is the magnitude of the total displacement?

$$\sqrt{12^2$ + 10^2$ + 2 \times 12 \times 10 \times \cos(150^\circ)}$$

Q: If $$\vec{C} = -2\hat{i} + 5\hat{j} m$$, what is the direction angle $$\theta of$$ $$\vec{C}$$ with respect to the positive x-axis?

$$\theta = \arctan\left(\frac{5}{-2}\right)$$

Question 1

Given the vectors in the first image, what is the canonical (Cartesian) representation of vector $$\vec{A}$$?

Accepted Answer

$$\vec{A} = 6\hat{i} + 5\hat{j}$$

Question 2

Using the first image, calculate the magnitude of the resultant vector $$\vec{R} = \vec{A} + \vec{B} + \vec{C} + \vec{D}.$$

Accepted Answer

$$\|\vec{R}\| = 5$$

Question 3

Given $$\vec{A} = 6\hat{i} - 3\hat{j}$$ and $$\vec{B} = -6\hat{i} - \hat{j}$$, what is $$2\vec{A} - \vec{B}$$?

Accepted Answer

$$18\hat{i} - 5\hat{j}$$

Question 4

From the second image, if $$\vec{A} = 4\hat{i} + 3\hat{j}$$ and $$\vec{B} = -3\hat{i} + 4\hat{j}$$, what is $$\vec{A} + \vec{B}$$?

Accepted Answer

$$1\hat{i} + 7\hat{j}$$

Question 5

Given the third image, what are the Cartesian components of $$\vec{B} (15.0 m$$, 30°)?

Accepted Answer

$$\vec{B} = 15.0\cos(30^\circ)\hat{i} + 15.0\sin(30^\circ)\hat{j}$$

Question 6

A physics professor drives 3.25 km north, 4.75 km west, and 1.50 km south. What is the magnitude of the resultant displacement?

Accepted Answer

$$\sqrt{(4.75)^2$ + (3.25-1.50)^2$} km$$

Question 7

Given $$\vec{A} = 2\hat{i} - 8\hat{j}$$ and $$\vec{B}$$ has magnitude 7 and forms an angle $$\alpha = 15^\circ$$ with the positive x-axis, what is the $$x-$$component of $$\vec{B}$$?

Accepted Answer

$$7\cos(15^\circ)$$

Question 8

If $$\vec{U} + \vec{V} + \vec{W} = 0$$ and $$\vec{U}$$ and $$\vec{V}$$ are known, what must be true about $$\vec{W}$$?

Accepted Answer

$$\vec{W} = - (\vec{U} + \vec{V})$$

Question 9

Given $$\vec{A} = \hat{i} + 3\hat{j} m$$, $$\vec{B} = \hat{i} - 4\hat{j} m$$, and $$\vec{C} = -2\hat{i} + 5\hat{j} m$$, what is $$\|\vec{A}\|$$?

Accepted Answer

$$\sqrt{1^2$ + 3^2$} m$$

Question 10

In the fourth image, if $$\|\vec{P}\| = 100$$, $$\|\vec{K}\| = 150$$, $$\|\vec{L}\| = 90$$, and $$\|\vec{N}\| = 120$$, which equation gives the $$x-$$component of $$\vec{R} = \vec{P} + \vec{K} + \vec{L} + \vec{N}$$?

Accepted Answer

$$R_x = P\cos\alpha + K\cos\beta + L\cos	heta + N\cos\mu$$

Question 11

A soccer player runs 12 m straight, then turns 30° to the left and runs 10 m. What is the magnitude of the total displacement?

Accepted Answer

$$\sqrt{12^2$ + 10^2$ + 2 	imes 12 	imes 10 	imes \cos(150^\circ)}$$

Question 12

Given $$\vec{A} = 6\hat{i} - 3\hat{j}$$ and $$\vec{B} = -6\hat{i} - \hat{j}$$, what is $$\vec{A} + \vec{B}$$?

Accepted Answer

$$0\hat{i} - 4\hat{j}$$

Question 13

If $$\vec{C} = -2\hat{i} + 5\hat{j} m$$, what is the direction angle $$	heta of$$ $$\vec{C}$$ with respect to the positive x-axis?

Accepted Answer

$$	heta = \arctan\left(\frac{5}{-2}ight)$$

Question 14

If $$\vec{A} = 2\hat{i} - 8\hat{j}$$ and $$\vec{B}$$ has magnitude 7 and forms an angle $$\alpha = 15^\circ$$ with the positive x-axis, what is $$\vec{C} = \frac{1}{2}\vec{A} + \vec{B}$$?

Accepted Answer

$$\vec{C} = (1 + 7\cos(15^\circ))\hat{i} + (-4 + 7\sin(15^\circ))\hat{j}$$

Question 15

Which of the following best describes the process to find the magnitude of a vector $$\vec{V} = a\hat{i} + b\hat{j}$$?

Accepted Answer

$$\|\vec{V}\| = \sqrt{a^2$ + b^2$}$$

Question 16

If $$\vec{R} = 2(\vec{A} + \vec{B}) - 3(\vec{D} + \vec{E}) + \vec{F}$$, which operation should be performed first according to the order of operations?

Accepted Answer

Calculate $$(\vec{A} + \vec{B})$$ and $$(\vec{D} + \vec{E})$$

Vector Operations and Representation – Physics with Calculus Study Guide

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