$$D_x = 10 \cos 30^\circ = 10 \times \frac{\sqrt{3}}{2} = 5\sqrt{3}$$

$$\theta = \tan$$^{-1}$$\left(\frac{24}{7}\right) \approx 73.7^\circ$$

Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

$$\vec{C} = (2 - 1)\hat{i} + (3 + 4)\hat{j} = \hat{i} + 7\hat{j}$$

Question 3

If a vector $$\vec{D}$$ has a magnitude of $$10$$ and makes an angle of $$30^\circ$$ with the $$x-$$axis, what is its $$x-$$component $$D_x$$?

Accepted Answer

$$D_x = 10 \cos 30^\circ = 10 	imes \frac{\sqrt{3}}{2} = 5\sqrt{3}$$

Question 4

Which of the following correctly expresses a vector $$\vec{A} in $$terms of its components and unit vectors?

Accepted Answer

$$\vec{A} = A_x \hat{i} + A_y \hat{j}$$

Question 5

A displacement vector in 2D has components $$x = 7 m $$and $$y = 24 m. $$What is the direction angle $$	heta$$ with respect to the $$x-$$axis?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{24}{7}ight) \approx 73.7^\circ$$

Question 6

Which statement about unit vectors is correct?

Accepted Answer

Unit vectors have a magnitude of $$1$$ and indicate direction along coordinate axes.

Question 7

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

Accepted Answer

$$\vec{A} - \vec{B} = (5 - (-3))\hat{i} + (-2 - 4)\hat{j} = 8\hat{i} - 6\hat{j}$$

Question 8

A vector $$\vec{F}$$ has components $$F_x = -6 N $$and $$F_y = 8 N. $$What is the magnitude of $$\vec{F}$$?

Accepted Answer

$$\sqrt{(-6)^2$ + 8^2$} = \sqrt{36 + 64} = 10 N$$

Question 9

Which of the following is a scalar quantity?

Accepted Answer

Temperature

Question 10

A vector $$\vec{R} is $$the sum of $$\vec{A} = 4\hat{i} + 2\hat{j}$$ and $$\vec{B} = -1\hat{i} + 5\hat{j}. $$What is the magnitude of $$\vec{R}$$?

Accepted Answer

$$\sqrt{(4 + (-1))^2$ + (2 + 5)^2$} = \sqrt{3^2$ + 7^2$} = \sqrt{58}$$

Question 11

A vector $$\vec{P}$$ has components $$P_x = 0$$ and $$P_y = 10. $$What is the direction of $$\vec{P}$$?

Accepted Answer

$$90^\circ$$ from the $$x-$$axis

Question 12

Which operation is used to find the component of a vector along a given axis?

Accepted Answer

Multiplying the magnitude by the cosine of the angle with the axis: $$A_x = |\vec{A}| \cos 	heta$$

Question 13

A vector $$\vec{Q} is $$described as the hypotenuse of a right triangle with legs $$Q_x$$ and $$Q_y. $$Which formula gives its magnitude?

Accepted Answer

$$|\vec{Q}| = \sqrt{Q_x^2$ + Q_y^2$}$$

Question 14

If a vector $$\vec{S}$$ has components $$S_x = 8$$ and $$S_y = 15$$, what is the magnitude of $$\vec{S}$$?

Accepted Answer

$$\sqrt{8^2$ + 15^2$} = \sqrt{64 + 225} = \sqrt{289} = 17$$

Question 15

A vector $$\vec{T}$$ has a magnitude of $$20$$ and points at $$60^\circ$$ above the $$x-$$axis. What is its $$y-$$component $$T_y$$?

Accepted Answer

$$T_y = 20 \sin 60^\circ = 20 	imes \frac{\sqrt{3}}{2} = 10\sqrt{3}$$

Question 16

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

Accepted Answer

Breaking the vector into perpendicular parts along the coordinate axes using trigonometric functions.

Question 17

A vector $$\vec{U}$$ has components $$U_x = -5$$ and $$U_y = 12. $$What is the angle $$	heta it $$makes with the $$x-$$axis?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{12}{-5}ight) \approx -67.4^\circ$$

Question 18

Which of the following is NOT a property of vectors?

Accepted Answer

Vectors can be divided by other vectors to yield another vector.

Vectors and Their Components in Physics

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