Q: A vector $$\vec{R} is $$the sum of $$\vec{A}$$ and $$\vec{B}. If$$ $$\vec{A} is $$along the $$x-$$axis and $$\vec{B} is $$along the $$y-$$axis, what is the angle $$\theta of$$ $$\vec{R}$$ with respect to the $$x-$$axis?

$$\theta = \arctan \left( \frac{B}{A} \right)$$

Question 1

Which of the following best describes the difference between distance and displacement as shown in the first image?

Accepted Answer

Distance is the total length of the path traveled, while displacement is the straight-line vector from the initial to the final position.

Question 2

A plane flies 200 miles at an angle of $$50^\circ$$ north of east. What are the $$x$$ and $$y$$ components of its displacement?

Accepted Answer

$$x = 200 \cos 50^\circ,\ y = 200 \sin 50^\circ$$

Question 3

Given vector $$\vec{A}$$ with magnitude $$12\ \mathrm{m} at$$ $$37^\circ$$ above the $$x-$$axis, what is $$A_x$$?

Accepted Answer

$$A_x = 12 \cos 37^\circ$$

Question 4

A vector $$\vec{B}$$ has a magnitude of $$15\ \mathrm{m}$$ and points $$40^\circ$$ below the $$x-$$axis. What is the sign of $$B_y$$?

Accepted Answer

Negative, because it points below the $$x-$$axis.

Question 5

Which quadrant does a vector point to if its $$x-$$component is negative and its $$y-$$component is positive?

Accepted Answer

Quadrant 2

Question 6

If $$\vec{A} = 12\ \mathrm{m} at$$ $$37^\circ$$ and $$\vec{B} = 15\ \mathrm{m} at$$ $$40^\circ$$ below the $$x-$$axis, what is the $$y-$$component of $$\vec{B}$$?

Accepted Answer

$$B_y = -15 \sin 40^\circ$$

Question 7

A hiker walks 60 km at $$70^\circ$$ north of east. What is the $$y-$$component of the displacement?

Accepted Answer

$$60 \sin 70^\circ$$

Question 8

Given $$\vec{C} = 6\ \mathrm{m} at$$ $$60^\circ$$ below the $$x-$$axis, what are the signs of $$C_x$$ and $$C_y$$?

Accepted Answer

$$C_x is $$positive, $$C_y is $$negative

Question 9

If $$\vec{A}$$ and $$\vec{B}$$ are vectors, which equation correctly represents the vector subtraction $$\vec{A} - \vec{B}$$?

Accepted Answer

$$\vec{A} - \vec{B} = \vec{A} + (-\vec{B})$$

Question 10

A boat travels 100 km east, then 200 km at $$50^\circ$$ north of east. What is the magnitude of the resultant displacement?

Accepted Answer

$$\sqrt{(100 + 200 \cos 50^\$$circ)^2$$ + (200 \sin 50^\$$circ)^2$$}$$

Question 11

Which of the following is the correct method to find the $$y-$$component of a vector $$\vec{A}$$ given its magnitude $$A$$ and angle $$	heta$$ from the $$x-$$axis?

Accepted Answer

$$A_y = A \sin 	heta$$

Question 12

A vector $$\vec{R} is $$the sum of $$\vec{A}$$ and $$\vec{B}. If$$ $$\vec{A} is $$along the $$x-$$axis and $$\vec{B} is $$along the $$y-$$axis, what is the angle $$	heta of$$ $$\vec{R}$$ with respect to the $$x-$$axis?

Accepted Answer

$$	heta = \arctan \left( \frac{B}{A} ight)$$

Question 13

A vector $$\vec{D}$$ points into the third quadrant. Which of the following is true about its components?

Accepted Answer

$$D_x \u003c 0,\ D_y \u003c 0$$

Question 14

A vector $$\vec{A}$$ has components $$A_x = 8$$ and $$A_y = 6. $$What is the magnitude of $$\vec{A}$$?

Accepted Answer

$$\sqrt{8^2$ + 6^2$}$$

Question 15

A vector $$\vec{B}$$ has $$B_x = 5$$ and $$B_y = -12. $$What is the direction of $$\vec{B}$$ with respect to the $$x-$$axis?

Accepted Answer

$$\arctan \left( \frac{-12}{5} \right)$$ below the $$x-$$axis

Models, Measurements, and Vectors: Foundations for Physics with Calculus

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