Q: A vector $$\vec{B}$$ has $$B_x = 8$$ and $$B_y = -6. $$What is the direction angle $$\theta ($$measured counterclockwise from the $$x-$$axis)?

$$\theta = \arctan\left(\frac{-6}{8}\right)$$

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 $$12\,\mathrm{m} at an $$angle of $$37^\circ$$ above the $$x-$$axis, what is $$A_x$$?

Accepted Answer

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

Question 4

If $$\vec{B}$$ 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

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

Accepted Answer

$$60 \sin 70^\circ$$

Question 7

Given $$\vec{A} = 100$$ units at $$0^\circ$$ and $$\vec{B} = 200$$ units at $$50^\circ$$, what is the magnitude of the resultant vector $$\vec{R}$$?

Accepted Answer

$$\sqrt{100^2$ + 200^2$ + 2 \cdot 100 \cdot 200 \cos 130^\circ}$$

Question 8

Which of the following is the correct method to find the $$y-$$component of a vector $$\vec{A} at an $$angle $$	heta$$ above the $$x-$$axis?

Accepted Answer

$$A_y = A \sin 	heta$$

Question 9

A boat sails $$62\,\mathrm{km} at$$ $$72.8^\circ$$ north of east. What is the $$x-$$component of its displacement?

Accepted Answer

$$62 \cos 72.8^\circ$$

Question 10

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 11

A vector $$\vec{C}$$ has a magnitude of $$6\,\mathrm{m}$$ and points $$60^\circ$$ 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 12

Which of the following best describes the process of resolving a vector into its components?

Accepted Answer

Projecting the vector onto the $$x$$ and $$y$$ axes to find perpendicular components.

Question 13

Given vectors $$\vec{A}$$ and $$\vec{B}$$, both in the first quadrant, which of the following is true about their $$x$$ and $$y$$ components?

Accepted Answer

Both $$B_x$$ and $$B_y$$ are positive.

Question 14

A vector $$\vec{D}$$ points into the third quadrant. Which of the following statements is correct?

Accepted Answer

Both $$D_x$$ and $$D_y$$ are negative.

Question 15

If $$\vec{A}$$ has components $$A_x = 9.6$$ and $$A_y = 7.2$$, what is the magnitude of $$\vec{A}$$?

Accepted Answer

$$\sqrt{9.6^2$ + 7.2^2$}$$

Question 16

A vector $$\vec{B}$$ has $$B_x = 8$$ and $$B_y = -6. $$What is the direction angle $$	heta ($$measured counterclockwise from the $$x-$$axis)?

Accepted Answer

$$	heta = \arctan\left(\frac{-6}{8}ight)$$

Question 17

A pilot flies $$45\,\mathrm{km}$$ west, then $$60\,\mathrm{km} at$$ $$70^\circ$$ south of east. What is the best method to find the resultant displacement?

Accepted Answer

Resolve each vector into $$x$$ and $$y$$ components, sum the components, then use the Pythagorean theorem.

Question 18

Which of the following is the correct formula for the magnitude of the resultant vector $$\vec{R}$$ given $$R_x$$ and $$R_y$$?

Accepted Answer

$$R = \sqrt{R_x^2$ + R_y^2$}$$

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

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