Q: In the sixteenth image, vector $$\vec{C} is $$shown. If $$\vec{C}$$ has components $$(-3, -1)$$, what is the direction of $$\vec{C}$$ measured counterclockwise from the positive $$x-$$axis?

$$\theta = \tan$$^{-1}$$\left(\frac{-1}{-3}\right) = 18.4^\circ$$ below the negative $$x-$$axis$

Question 1

A ball is thrown at an angle above the horizontal and follows a parabolic path, as shown in the first image. Which vector quantity is always directed downward throughout the motion?

Accepted Answer

Acceleration

Question 2

In the second image, the coordinates of point $$P$$ are shown as $$(P_x, P_y). If$$ $$P_x = 3$$ and $$P_y = 4$$, what is the distance from the origin to point $$P$$?

Accepted Answer

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

Question 3

Refer to the third image. What is the coordinate of the red point $$P_x on $$the x-axis?

Accepted Answer

$$-2$$

Question 4

In the fourth image, a point $$P is $$shown in three-dimensional space. Which of the following correctly represents its coordinates?

Accepted Answer

$$(P_x, P_y, P_z)$$

Question 5

In the fifth image, vector $$\vec{A}$$ points from $$P to$$ $$Q. If$$ $$P = (-2, -2)$$ and $$Q = (2, 2)$$, what are the components of $$\vec{A}$$?

Accepted Answer

$$(4, 4)$$

Question 6

The sixth image shows unit vectors $$\hat{x}$$ and $$\hat{y}. $$What is the magnitude of each unit vector?

Accepted Answer

$$1$$

Question 7

In the seventh image, vector $$\vec{A} is $$shown with components $$A_x$$ and $$A_y. If$$ $$|\vec{A}| = 10$$ and $$	heta = 30^\circ$$, what is $$A_x$$?

Accepted Answer

$$A_x = 10 \cos 30^\circ$$

Question 8

In the eighth image, the vector $$\vec{a}$$ has a magnitude of $$6.0\ \mathrm{m/s^2$}$$ and is at an angle of $$30^\circ$$ below the horizontal. What is the $$x-$$component of $$\vec{a}$$?

Accepted Answer

$$a_x = 6.0 \cos 30^\circ$$

Question 9

In the ninth image, the vector $$\vec{C} is $$shown. If the tail is at $$(1, -1)$$ and the head is at $$(-3, 2)$$, what are the components of $$\vec{C}$$?

Accepted Answer

$$(-4, 3)$$

Question 10

In the tenth image, what is the magnitude of vector $$\vec{C}$$ with components $$(-4, 3)$$?

Accepted Answer

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

Question 11

In the eleventh image, which vector operation is illustrated by the parallelogram law?

Accepted Answer

Vector addition

Question 12

In the twelfth image, vectors $$\vec{A}$$ and $$\vec{B}$$ are added using components. If $$A_x = 3$$, $$A_y = 2$$, $$B_x = 4$$, and $$B_y = 5$$, what are the components of $$\vec{C} = \vec{A} + \vec{B}$$?

Accepted Answer

$$(7, 7)$$

Question 13

The thirteenth image shows $$\vec{C}$$ and $$-\vec{C}. $$What is the result of $$\vec{C} + (-\vec{C})$$?

Accepted Answer

$$0$$

Question 14

In the fourteenth image, the vector $$\vec{E} is $$expressed as $$E_x \hat{i} + E_y \hat{j}. If$$ $$E_x = 5$$ and $$E_y = 12$$, what is the magnitude of $$\vec{E}$$?

Accepted Answer

$$\sqrt{5^2$ + 12^2$} = 13$$

Question 15

In the fifteenth image, which of the following statements is true about unit vectors $$\hat{i}$$ and $$\hat{j}$$?

Accepted Answer

They have magnitude 1 and point along the $$x$$ and $$y$$ axes, respectively.

Question 16

In the sixteenth image, vector $$\vec{C} is $$shown. If $$\vec{C}$$ has components $$(-3, -1)$$, what is the direction of $$\vec{C}$$ measured counterclockwise from the positive $$x-$$axis?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{-1}{-3}ight) = 18.4^\circ$$ below the negative $$x-$$axis$

Question 17

In the seventeenth image, which vector represents $$\vec{e} - 2\vec{a} if $$each square is one unit?

Accepted Answer

$$(-3, 4)$$

Question 18

In the eighteenth image, the coordinate axes are rotated by $$30^\circ. $$What is the $$x-$$component of a vector $$\vec{A} of $$magnitude $$A$$ along the new $$x-$$axis?

Accepted Answer

$$A \cos 30^\circ$$

Physics with Calculus: Motion, Vectors, and Coordinate Systems

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