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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