Q: A hiker walks 3 miles east and then 4 miles north. What is the magnitude of the hiker's net displacement?

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

Q: If a bird flies 100 m due east and then 50 m northwest (45° north of west), what is the magnitude of its net displacement?

$$\sqrt{(100)^2$ + (50)^2$ - 2 \times 100 \times 50 \times \cos 45^\circ} = 74\ \text{m}$$

Q: A rabbit runs at $$10.0\ \text{m/s}$$, $$40.0^\circ$$ north of west. What is the x-component of its velocity?

$$-10.0 \times \cos 40.0^\circ = -7.66\ \text{m/s}$$

Q: A vector $$\vec{B}$$ has components $$B_x = -3\ \text{cm}$$ and $$B_y = 1\ \text{cm}. $$What is the direction angle $$\theta$$ measured from the positive x-axis?

$$\theta = \tan$$^{-1}$$(1 / -3)$$

Q: A vector $$\vec{C}$$ has components $$C_x = 64.7\ \text{m}$$ and $$C_y = 35.3\ \text{m}. $$What is the magnitude of $$\vec{C}$$?

$$\sqrt{(64.7)^2$ + (35.3)^2$} = 74\ \text{m}$$

Q: Which equation correctly gives the direction angle $$\phi of a $$vector $$\vec{C}$$ with components $$C_x$$ and $$C_y$$?

$$\phi = \tan$$^{-1}$$(C_y / C_x)$$

Question 1

Which of the following is a correct definition of a vector quantity?

Accepted Answer

A quantity that has both magnitude and direction.

Question 2

Sam walks 200 ft northeast from his front door. What is the magnitude of his displacement vector $$\vec{S}$$?

Accepted Answer

$$\vec{S} = 200\ 	ext{ft}$$

Question 3

A hiker walks 3 miles east and then 4 miles north. What is the magnitude of the hiker's net displacement?

Accepted Answer

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

Question 4

If a bird flies 100 m due east and then 50 m northwest (45° north of west), what is the magnitude of its net displacement?

Accepted Answer

$$\sqrt{(100)^2$ + (50)^2$ - 2 	imes 100 	imes 50 	imes \cos 45^\circ} = 74\ 	ext{m}$$

Question 5

Which rule allows you to find the sum of two vectors by drawing them with their tails together and constructing the diagonal?

Accepted Answer

Parallelogram rule

Question 6

Given vectors $$\vec{A}$$ and $$\vec{B}$$, which equation correctly represents their vector sum using components?

Accepted Answer

$$\vec{C} = (A_x + B_x)\hat{i} + (A_y + B_y)\hat{j}$$

Question 7

A rabbit runs at $$10.0\ 	ext{m/s}$$, $$40.0^\circ$$ north of west. What is the x-component of its velocity?

Accepted Answer

$$-10.0 	imes \cos 40.0^\circ = -7.66\ 	ext{m/s}$$

Question 8

Which of the following is a unit vector in the positive y-direction?

Accepted Answer

$$\hat{j}$$

Question 9

If $$\vec{A} = 3\hat{i} + 4\hat{j}$$, what is the magnitude of $$\vec{A}$$?

Accepted Answer

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

Question 10

Which of the following statements about coordinate systems is correct?

Accepted Answer

Axes can be oriented arbitrarily as long as they are perpendicular.

Question 11

A vector $$\vec{B}$$ has components $$B_x = -3\ 	ext{cm}$$ and $$B_y = 1\ 	ext{cm}. $$What is the direction angle $$	heta$$ measured from the positive x-axis?

Accepted Answer

$$	heta = 	an$$^{-1}$$(1 / -3)$$

Question 12

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

$$2\hat{i} + 2\hat{j}$$

Question 13

A vector $$\vec{C}$$ has components $$C_x = 64.7\ 	ext{m}$$ and $$C_y = 35.3\ 	ext{m}. $$What is the magnitude of $$\vec{C}$$?

Accepted Answer

$$\sqrt{(64.7)^2$ + (35.3)^2$} = 74\ 	ext{m}$$

Question 14

Which of the following is NOT a property of the zero vector?

Accepted Answer

It is parallel to any vector.

Question 15

If a vector $$\vec{A}$$ points downwards, which component will be negative in a standard xy-coordinate system?

Accepted Answer

The y-component $$A_y$$

Question 16

Which equation correctly gives the direction angle $$\phi of a $$vector $$\vec{C}$$ with components $$C_x$$ and $$C_y$$?

Accepted Answer

$$\phi = 	an$$^{-1}$$(C_y / C_x)$$

Question 17

A GPS system uses coordinate axes to determine your position. Which statement best describes the role of coordinate systems in physics?

Accepted Answer

Coordinate systems are arbitrary grids imposed to simplify analysis.

Question 18

If $$\vec{A} = 4\hat{i} - 3\hat{j}$$, what is the direction angle $$	heta$$ measured from the positive x-axis?

Accepted Answer

$$	heta = 	an$$^{-1}$$(-3 / 4)$$

Vectors and Coordinate Systems – Study Notes

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