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

$$\theta = \tan$$^{-1}$$\left(\frac{8}{-6}\right) = 126.9^\circ$$

Q: Which trigonometric function relates the opposite side and the hypotenuse in a right triangle?

$$\sin \theta = \frac{\text{opp}}{\text{hyp}}$$

Question 1

A vector $$\vec{A}$$ has a magnitude of 10 units and makes an angle of $$60^\circ$$ with the positive x-axis. What are the x and y components of $$\vec{A}$$?

Accepted Answer

$$A_x = 10 \cos 60^\circ = 5$$, $$A_y = 10 \sin 60^\circ = 8.66$$

Question 2

Given two vectors $$\vec{A} = \langle 3, 4 \rangle$$ and $$\vec{B} = \langle -2, 0 \rangle$$, what is the magnitude of their sum $$\vec{R} = \vec{A} + \vec{B}$$?

Accepted Answer

$$\sqrt{(3 + (-2))^2$ + (4 + 0)^2$} = \sqrt{1^2$ + 4^2$} = \sqrt{17} \approx 4.12$$

Question 3

A plane is flying at $$100\ \mathrm{km/h}$$, $$30^\circ$$ north of east. The wind is blowing at $$20\ \mathrm{km/h}$$, $$45^\circ$$ south of west. What is the x-component of the plane's velocity relative to the ground?

Accepted Answer

$$v_{pg,x} = 100 \cos 30^\circ + 20 \cos 225^\circ = 86.6 - 14.1 = 72.5\ \mathrm{km/h}$$

Question 4

Which equation correctly gives the magnitude of a vector $$\vec{A} in $$three dimensions with components $$A_x$$, $$A_y$$, and $$A_z$$?

Accepted Answer

$$A = \sqrt{A_x^2$ + A_y^2$ + A_z^2$}$$

Question 5

If a vector $$\vec{C}$$ lies in the third quadrant of the xy-plane, which of the following statements is true about its components?

Accepted Answer

Both $$C_x$$ and $$C_y$$ are negative.

Question 6

Raoul walks 500 m at $$35^\circ$$ north of east. What is his displacement in the x-direction (east)?

Accepted Answer

$$500 \cos 35^\circ = 409.6\ \mathrm{m}$$

Question 7

Which of the following best describes the process for adding two vectors $$\vec{A}$$ and $$\vec{B}$$ using components?

Accepted Answer

Add the x-components and y-components separately: $$R_x = A_x + B_x$$, $$R_y = A_y + B_y.$$

Question 8

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

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{8}{-6}ight) = 126.9^\circ$$

Question 9

Maria walks 700 m at $$55^\circ$$ south of west. What is her displacement in the y-direction (south)?

Accepted Answer

$$700 \sin 55^\circ = 573.6\ \mathrm{m} ($$negative y-direction)

Question 10

Which trigonometric function relates the opposite side and the hypotenuse in a right triangle?

Accepted Answer

$$\sin 	heta = \frac{	ext{opp}}{	ext{hyp}}$$

Question 11

A vector $$\vec{A}$$ has components $$A_x = 4$$ and $$A_y = 3. $$What is the magnitude of $$\vec{A}$$?

Accepted Answer

$$\sqrt{4^2$ + 3^2$} = 5$$

Question 12

If $$A_x = A \cos 	heta$$ and $$A_y = A \sin 	heta$$, what is $$	heta if$$ $$A_x = 0$$ and $$A_y = -A$$?

Accepted Answer

$$	heta = 270^\circ$$

Question 13

Which of the following is a correct statement about vector addition using the tip-to-tail method?

Accepted Answer

The tail of the second vector is placed at the tip of the first vector.

Question 14

A vector $$\vec{D}$$ has a magnitude of 2 units and points directly west. What are its components?

Accepted Answer

$$D_x = -2$$, $$D_y = 0$$

Question 15

If $$\vec{A}$$ and $$\vec{B}$$ are perpendicular vectors, which equation gives the magnitude of their sum $$\vec{R}$$?

Accepted Answer

$$R = \sqrt{A^2$ + B^2$}$$

Question 16

A vector $$\vec{A}$$ has a magnitude $$A$$ and makes an angle $$	heta$$ with the positive x-axis. Which equation gives the y-component?

Accepted Answer

$$A_y = A \sin 	heta$$

Study Notes: Vectors and Their Components in Physics

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