Q: Given the position equation $$x = A + Bt + Ct^2$, what are the dimensions of the constant C$?

$$[C] = \text{distance}/\text{time}^2$$

Q: If a vector has a magnitude of $$75.0\ \text{cm}$$ and points $$30^\circ$$ above the positive x-axis, what is its y-component?

$$y = 75.0 \times \sin(30^\circ) = 38\ \text{cm}$$

Q: A vector of $$25.0\ \text{cm}$$ points along the negative x-axis. What are its x and y components?

$$x = -25.0\ \text{cm},\ y = 0$$

Q: A vector of $$40.0\ \text{cm}$$ points $$45^\circ$$ below the negative x-axis. What is its y-component?

$$y = -40.0 \times \sin(45^\circ) = -28\ \text{cm}$$

Q: Three vectors have x-components $$65\ \text{cm}$$, $$-25\ \text{cm}$$, and $$-28\ \text{cm}$$, and y-components $$38\ \text{cm}$$, $$0$$, and $$-28\ \text{cm}$$ respectively. What is the x-component of their sum?

$$x_{\text{sum}} = 65 - 25 - 28 = 12\ \text{cm}$$

Q: Given the sum of three vectors with x-component $$12\ \text{cm}$$ and y-component $$9.2\ \text{cm}$$, what is the magnitude of the resultant vector?

$$\sqrt{12^2$ + 9.2^2$} = 15\ \text{cm}$$

Q: A vector sum has x-component $$12\ \text{cm}$$ and y-component $$9.2\ \text{cm}. $$What is the direction of the resultant vector relative to the positive x-axis?

$$\theta = \tan$$^{-1}$$\left(\frac{9.2}{12}\right) = 38^\circ$$

Q: A vector is $$7.5\ \text{cm}$$ long and points at $$+30^\circ to $$the negative x-axis. What is its x-component?

$$x = -7.5 \times \cos(30^\circ) = -6.5\ \text{cm}$$

Q: Two vectors have components $$x_1 = 5.5$$\ \text{cm}$$, y_1$ = 0$$ and $$x_2 = -6.5$$\ \text{cm}$$, y_2$ = 3.8\ \text{cm}. $$What are the components of their sum?

$$x = -1.0\ \text{cm},\ y = 3.8\ \text{cm}$$

Q: Given the sum of two vectors with components $$x = -1.0\ \text{cm}$$ and $$y = 3.8\ \text{cm}$$, what is the magnitude of the resultant vector?

$$\sqrt{(-1.0)^2$ + (3.8)^2$} = 3.9\ \text{cm}$$

Question 1

Given the position equation $$x = A + Bt + Ct^2$, what are the dimensions of the constant C$?

Accepted Answer

$$[C] = 	ext{distance}/	ext{time}^2$$

Question 2

If a vector has a magnitude of $$75.0\ 	ext{cm}$$ and points $$30^\circ$$ above the positive x-axis, what is its y-component?

Accepted Answer

$$y = 75.0 	imes \sin(30^\circ) = 38\ 	ext{cm}$$

Question 3

A vector of $$25.0\ 	ext{cm}$$ points along the negative x-axis. What are its x and y components?

Accepted Answer

$$x = -25.0\ 	ext{cm},\ y = 0$$

Question 4

A vector of $$40.0\ 	ext{cm}$$ points $$45^\circ$$ below the negative x-axis. What is its y-component?

Accepted Answer

$$y = -40.0 	imes \sin(45^\circ) = -28\ 	ext{cm}$$

Question 5

Three vectors have x-components $$65\ 	ext{cm}$$, $$-25\ 	ext{cm}$$, and $$-28\ 	ext{cm}$$, and y-components $$38\ 	ext{cm}$$, $$0$$, and $$-28\ 	ext{cm}$$ respectively. What is the x-component of their sum?

Accepted Answer

$$x_{	ext{sum}} = 65 - 25 - 28 = 12\ 	ext{cm}$$

Question 6

Given the sum of three vectors with x-component $$12\ 	ext{cm}$$ and y-component $$9.2\ 	ext{cm}$$, what is the magnitude of the resultant vector?

Accepted Answer

$$\sqrt{12^2$ + 9.2^2$} = 15\ 	ext{cm}$$

Question 7

A vector sum has x-component $$12\ 	ext{cm}$$ and y-component $$9.2\ 	ext{cm}. $$What is the direction of the resultant vector relative to the positive x-axis?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{9.2}{12}ight) = 38^\circ$$

Question 8

A vector is $$5.5\ 	ext{cm}$$ long and points along the x-axis. What are its x and y components?

Accepted Answer

$$x = 5.5\ 	ext{cm},\ y = 0$$

Question 9

A vector is $$7.5\ 	ext{cm}$$ long and points at $$+30^\circ to $$the negative x-axis. What is its x-component?

Accepted Answer

$$x = -7.5 	imes \cos(30^\circ) = -6.5\ 	ext{cm}$$

Question 10

Two vectors have components $$x_1 = 5.5$$\ 	ext{cm}$$, y_1$ = 0$$ and $$x_2 = -6.5$$\ 	ext{cm}$$, y_2$ = 3.8\ 	ext{cm}. $$What are the components of their sum?

Accepted Answer

$$x = -1.0\ 	ext{cm},\ y = 3.8\ 	ext{cm}$$

Question 11

Given the sum of two vectors with components $$x = -1.0\ 	ext{cm}$$ and $$y = 3.8\ 	ext{cm}$$, what is the magnitude of the resultant vector?

Accepted Answer

$$\sqrt{(-1.0)^2$ + (3.8)^2$} = 3.9\ 	ext{cm}$$

Question 12

What is the direction of the vector sum with components $$x = -1.0\ 	ext{cm}$$ and $$y = 3.8\ 	ext{cm}$$ relative to the negative x-axis?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{3.8}{-1.0}ight) = 75^\circ$$ above -x axis

Question 13

Given vector $$\vec{A} = 3.00\hat{i} + 1.00\hat{j}$$ and another vector $$\vec{B}$$, what is the angle between their directions if the answer is $$117^\circ$$?

Accepted Answer

$$117^\circ$$

Question 14

If the magnitude of vector $$\vec{A} is$$ $$18.0$$ units and vector $$\vec{B} is$$ $$12.0$$ units, what must the magnitude and direction of vector $$\vec{C} be so $$that the resultant points in the $$-x$$ direction with magnitude $$7.50$$ units?

Accepted Answer

$$15.5$$ units at $$209^\circ$$

Question 15

If the position of an object is given by $$x = A + Bt + Ct^2$, which term represents the initial position?

Accepted Answer

$$A$$

Question 16

If a vector points $$45^\circ$$ below the negative x-axis, which quadrant does it lie in?

Accepted Answer

Third quadrant

Question 17

If the dimension of $$b is$$ $$[L/T]$$ and the dimension of $$c is$$ $$[L]$$, what is the dimension of $$a in $$the equation $$a = b \cdot c$$?

Accepted Answer

$$[a] = [L/T] \cdot [L] = [L^2$/T]$$

Question 18

A vector has components $$x = 12\ 	ext{cm}$$ and $$y = 9.2\ 	ext{cm}. $$What is the direction of the vector relative to the positive x-axis?

Accepted Answer

$$	heta = 	an$$^{-1}$$\left(\frac{9.2}{12}ight) = 38^\circ$$

Chapter 1 Physics Practice: Dimensional Analysis and Vectors

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