Q: Which trigonometric function would you use to find the angle $$\theta if $$you know the lengths of the opposite side and the hypotenuse in a right triangle?

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

Q: If you are given the x and y components of a vector, $$A_x$$ and $$A_y$$, how do you find the direction angle $$\theta of $$the vector relative to the x-axis?

$$\theta = \tan$$^{-1}$$ \left( \dfrac{A_y}{A_x} \right)$$

Q: If a vector has components $$A_x = -6\ \mathrm{m}$$ and $$A_y = 8\ \mathrm{m}$$, what is the direction angle $$\theta ($$relative to the x-axis)?

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

Question 1

Which of the following is a vector quantity?

Accepted Answer

6 m/s southwest

Question 2

A student is asked to resolve a vector of magnitude $$10\ \mathrm{m} at an $$angle of $$30^\circ$$ above the x-axis into its x and y components. Which of the following expressions correctly gives the y-component?

Accepted Answer

$$10 \sin 30^\circ$$

Question 3

Which equation correctly represents the Pythagorean Theorem as used in vector addition for perpendicular vectors $$\vec{A}$$ and $$\vec{B}$$?

Accepted Answer

$$a^2$$ + $$b^2$$ = $$c^2$$

Question 4

If a vector $$\vec{A}$$ has components $$A_x = 3\ \mathrm{m}$$ and $$A_y = 4\ \mathrm{m}$$, what is the magnitude of $$\vec{A}$$?

Accepted Answer

$$5\ \mathrm{m}$$

Question 5

Which trigonometric function would you use to find the angle $$	heta if $$you know the lengths of the opposite side and the hypotenuse in a right triangle?

Accepted Answer

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

Question 6

A vector $$\vec{B}$$ has a magnitude of $$12\ \mathrm{m}$$ and points $$37^\circ$$ above the x-axis. What is the x-component of $$\vec{B}$$?

Accepted Answer

$$12 \cos 37^\circ$$

Question 7

When adding two vectors that are perpendicular to each other, which method should you use to find the magnitude of the resultant vector?

Accepted Answer

Use the Pythagorean Theorem: $$C = \sqrt{A^2$ + B^2$}$$

Question 8

If you are given the x and y components of a vector, $$A_x$$ and $$A_y$$, how do you find the direction angle $$	heta of $$the vector relative to the x-axis?

Accepted Answer

$$	heta = 	an$$^{-1}$$ \left( \dfrac{A_y}{A_x} ight)$$

Question 9

Which of the following statements about scalar and vector quantities is correct?

Accepted Answer

Scalars have only magnitude, while vectors have both magnitude and direction

Question 10

A displacement vector is described as $$8\ \mathrm{m} at$$ $$60^\circ$$ above the x-axis. What is the y-component of this vector?

Accepted Answer

$$8 \sin 60^\circ$$

Question 11

If two vectors $$\vec{A}$$ and $$\vec{B}$$ are not parallel or perpendicular, what is the recommended method for finding their resultant?

Accepted Answer

Use the component method to add their x and y components

Question 12

A vector $$\vec{C}$$ has components $$C_x = -5\ \mathrm{m}$$ and $$C_y = 12\ \mathrm{m}. $$What is the magnitude of $$\vec{C}$$?

Accepted Answer

$$13\ \mathrm{m}$$

Question 13

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

Accepted Answer

Breaking a vector into perpendicular parts, usually along the x and y axes

Question 14

A vector $$\vec{A} is$$ $$10\ \mathrm{m} at$$ $$45^\circ$$ above the x-axis. What are the x and y components of $$\vec{A}$$?

Accepted Answer

$$A_x = 10 \cos 45^\circ$$, $$A_y = 10 \sin 45^\circ$$

Question 15

Which of the following is a common mistake when resolving vectors into components?

Accepted Answer

Assuming the x-component always uses the cosine function

Question 16

If a vector $$\vec{A} is$$ $$5\ \mathrm{m} in $$the positive x-direction and a vector $$\vec{B} is$$ $$12\ \mathrm{m} in $$the positive y-direction, what is the magnitude of the resultant vector $$\vec{C} = \vec{A} + \vec{B}$$?

Accepted Answer

$$13\ \mathrm{m}$$

Question 17

Which quadrant will the angle $$	heta be in if $$both the x and y components of a vector are negative?

Accepted Answer

Quadrant III

Question 18

A vector $$\vec{D}$$ has a magnitude of $$20\ \mathrm{m}$$ and points $$120^\circ$$ counterclockwise from the positive x-axis. What is the x-component of $$\vec{D}$$?

Accepted Answer

$$20 \cos 120^\circ$$

Question 19

When adding vectors that are not parallel or perpendicular, what is the first step in the component method?

Accepted Answer

Resolve all vectors into their x and y components

Question 20

If a vector has components $$A_x = -6\ \mathrm{m}$$ and $$A_y = 8\ \mathrm{m}$$, what is the direction angle $$	heta ($$relative to the x-axis)?

Accepted Answer

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

Vectors and Trigonometry in Physics: Structured Study Notes

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