Q: Given a vector $$\vec{v}$$ with components $$v_x = 6$$ and $$v_y = 8$$, what is the direction angle $$\theta of $$the vector from the positive x-axis?

$$\theta = \arctan\left(\frac{8}{6}\right) \approx 53.13^\circ$$

Q: A plane flies with a velocity vector of $$\vec{v} = (15, -20). $$What is the direction angle of the plane's velocity from the positive x-axis?

$$\theta = \arctan\left(\frac{-20}{15}\right) \approx -53.13^\circ$$

Q: A vector $$\vec{c}$$ has components $$c_x = 12$$ and $$c_y = -5. $$What is the direction angle $$\theta$$ from the positive x-axis?

$$\theta = \arctan\left(\frac{-5}{12}\right) \approx -22.62^\circ$$

Q: A vector $$\vec{e}$$ has components $$e_x = -9$$ and $$e_y = -12. $$What is the direction angle $$\theta$$ from the positive x-axis?

$$\theta = \arctan\left(\frac{-12}{-9}\right) + 180^\circ \approx 233.13^\circ$$

Question 1

Given a vector $$\vec{v}$$ with components $$v_x = 6$$ and $$v_y = 8$$, what is the direction angle $$	heta of $$the vector from the positive x-axis?

Accepted Answer

$$	heta = \arctan\left(\frac{8}{6}ight) \approx 53.13^\circ$$

Question 2

A vector $$\vec{u}$$ has a magnitude of $$10$$ and a direction of $$60^\circ$$ from the positive x-axis. What are its horizontal ($$u_x) $$and vertical ($$u_y) $$components?

Accepted Answer

$$u_x = 10 \cos 60^\circ = 5$$, $$u_y = 10 \sin 60^\circ \approx 8.66$$

Question 3

If a vector $$\vec{a}$$ has components $$a_x = -7$$ and $$a_y = 24$$, what is its magnitude?

Accepted Answer

$$|\vec{a}| = \sqrt{(-7)^2$ + 24^2$} = 25$$

Question 4

A plane flies with a velocity vector of $$\vec{v} = (15, -20). $$What is the direction angle of the plane's velocity from the positive x-axis?

Accepted Answer

$$	heta = \arctan\left(\frac{-20}{15}ight) \approx -53.13^\circ$$

Question 5

A vector $$\vec{b}$$ has a magnitude of $$5$$ and a direction of $$135^\circ. $$What is the value of its $$x-$$component?

Accepted Answer

$$b_x = 5 \cos 135^\circ \approx -3.54$$

Question 6

A vector $$\vec{c}$$ has components $$c_x = 12$$ and $$c_y = -5. $$What is the direction angle $$	heta$$ from the positive x-axis?

Accepted Answer

$$	heta = \arctan\left(\frac{-5}{12}ight) \approx -22.62^\circ$$

Question 7

A vector $$\vec{d}$$ has a magnitude of $$8$$ and a direction of $$210^\circ. $$What is the $$y-$$component of the vector?

Accepted Answer

$$d_y = 8 \sin 210^\circ = -4$$

Question 8

A vector $$\vec{e}$$ has components $$e_x = -9$$ and $$e_y = -12. $$What is the direction angle $$	heta$$ from the positive x-axis?

Accepted Answer

$$	heta = \arctan\left(\frac{-12}{-9}ight) + 180^\circ \approx 233.13^\circ$$

Question 9

A vector $$\vec{f}$$ has a magnitude of $$7$$ and a direction of $$300^\circ. $$What are its $$x$$ and $$y$$ components?

Accepted Answer

$$f_x = 7 \cos 300^\circ = 3.5$$, $$f_y = 7 \sin 300^\circ = -6.06$$

Question 10

A boat travels with a velocity vector $$\vec{v} = (20, 21). $$What is the magnitude of the velocity?

Accepted Answer

$$|\vec{v}| = \sqrt{20^2$ + 21^2$} = 29$$

Question 11

A vector $$\vec{g}$$ has a magnitude of $$12$$ and a direction of $$75^\circ. $$What is the $$y-$$component of the vector?

Accepted Answer

$$g_y = 12 \sin 75^\circ \approx 11.59$$

Question 12

A vector $$\vec{h}$$ has components $$h_x = 0$$ and $$h_y = 10. $$What is the direction angle from the positive x-axis?

Accepted Answer

$$	heta = 90^\circ$$

Question 13

A vector $$\vec{i}$$ has a magnitude of $$9$$ and a direction of $$270^\circ. $$What are its $$x$$ and $$y$$ components?

Accepted Answer

$$i_x = 9 \cos 270^\circ = 0$$, $$i_y = 9 \sin 270^\circ = -9$$

Question 14

A vector $$\vec{j}$$ has components $$j_x = 5$$ and $$j_y = 5. $$What is the direction angle from the positive x-axis?

Accepted Answer

$$	heta = \arctan\left(\frac{5}{5}ight) = 45^\circ$$

Question 15

A vector $$\vec{k}$$ has a magnitude of $$15$$ and a direction of $$225^\circ. $$What is the $$x-$$component of the vector?

Accepted Answer

$$k_x = 15 \cos 225^\circ \approx -10.61$$

Question 16

A vector $$\vec{l}$$ has components $$l_x = 3$$ and $$l_y = -4. $$What is the magnitude of the vector?

Accepted Answer

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

Question 17

A vector $$\vec{m}$$ has a magnitude of $$20$$ and a direction of $$120^\circ. $$What is the $$y-$$component of the vector?

Accepted Answer

$$m_y = 20 \sin 120^\circ \approx 17.32$$

Direction and Components of Vectors: Trigonometric Applications

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