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Precalculus : Matrices and Vectors

Study concepts, example questions & explanations for Precalculus

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12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Find A Direction Vector When Given Two Points

Find the direction vector that has an initial point at E(4, 1) and a terminal point at F(-5, -1) .

Possible Answers:

<-2, -9>

<-9, -2>

<9, 2>

<2, 9>

Correct answer:

<-9, -2>

Explanation:

To find the directional vector, subtract the coordinates of the initial point from the coordinates of the terminal point.

$\overrightarrow{EF}=<-5-4, -1-1>=<-9, -2>$

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Example Question #1 : Find A Direction Vector When Given Two Points

Find the direction vector with an initial point of R(-1, 4) and an terminal point of S(4, 9) .

Possible Answers:

<5,5>

<5, -5>

<-5, -5>

<-5, 5>

Correct answer:

<5,5>

Explanation:

To find the directional vector, subtract the coordinates of the initial point from the coordinates of the terminal point.

$\overrightarrow{RS}=<4-(-1), 9-4>=<5, 5>$

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Example Question #1 : Find A Direction Vector When Given Two Points

Find the direction vector that has an initial point at Z(1, 1) and a terminal point at A(4, 8) .

Possible Answers:

<-3, -7>

<3, 7>

<-3, 7>

<7, 3>

Correct answer:

<3, 7>

Explanation:

To find the directional vector, subtract the coordinates of the initial point from the coordinates of the terminal point.

$\overrightarrow{ZA}=<4-1, 8-1>=<3, 7>$

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Example Question #101 : Matrices And Vectors

Find the direction vector with an initial point of T(-4, 5) and a terminal point B(4, -1) .

Possible Answers:

<-8, 6>

<8, -6>

<-6, 8>

<-8, 6>

Correct answer:

<8, -6>

Explanation:

To find the directional vector, subtract the coordinates of the initial point from the coordinates of the terminal point.

$\overrightarrow{TB}=<4-(-4), -1-5>=<8, -6>$

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Example Question #12 : Find A Direction Vector When Given Two Points

If $\overrightarrow{DE}=<5, -1>$ and E=(-2, -1) , which of the following is a possible point for ?

Possible Answers:

(0, -3)

(-3, -2)

(-7,0)

(3, -2)

Correct answer:

(-7,0)

Explanation:

To find the directional vector, subtract the coordinates of the initial point from the coordinates of the terminal point.

To find the x-coordinate of ,

-2-x=5

x=-7

To find the y-coordinate of ,

-1-y=-1

y=0

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Example Question #11 : Find A Direction Vector When Given Two Points

If $\overrightarrow{EF}=<2, -8>$ and E=(-1, 5) , what is one possible point for ?

Possible Answers:

(1, -3)

(-3, -3)

(3, 3)

(-3, 3)

Correct answer:

(1, -3)

Explanation:

To find the directional vector, subtract the coordinates of the initial point from the coordinates of the terminal point.

To find the x-coordinate of ,

x-(-1)=2

x=1

To find the y-coordinate of ,

y-5=-8

y=-3

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Example Question #12 : Find A Direction Vector When Given Two Points

If $\overrightarrow{OP}=<4, -2>$ and O=(2, 10) , what is a possible value of ?

Possible Answers:

(-6, 8)

(6, -2)

(6, 8)

(8, 6)

Correct answer:

(6, 8)

Explanation:

To find the directional vector, subtract the coordinates of the initial point from the coordinates of the terminal point.

To find the x-coordinate of ,

x-2=4

x=6

To find the y-coordinate of ,

y-10=-2

y=8

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Example Question #1 : Express A Vector In Polar Form

Rewrite the vector $\vec{v}= 3 \hat{i} + 5 \hat{j}$ from Cartesian coordinates to polar coordinates $\vec{v} = \left \langle r \angle \theta \right \rangle$ .

Possible Answers:

$\vec{v} = \left \langle \sqrt{21} \angle 32.64^\circ \right \rangle$

$\vec{v} = \left \langle \sqrt{8} \angle 60^\circ \right \rangle$

$\vec{v} = \left \langle 16 \angle 60^\circ \right \rangle$

$\vec{v} = \left \langle \sqrt{34} \angle 59.04^\circ \right \rangle$

Correct answer:

$\vec{v} = \left \langle \sqrt{34} \angle 59.04^\circ \right \rangle$

Explanation:

To convert to polar form, we need to find the magnitude of the vector, , and the angle it forms with the positive -axis going counterclockwise, or $\theta$ . This is shown in the figure below.

Vecfssec

We find the angle using trigonometric identities:

$\tan{\theta}= \frac{5}{3}$

Using a calculator,

$\theta = \arctan{\frac{5}{3}} \approx 59.04^\circ$

To find the magnitude of a vector, we add up the squares of each component and take the square root:

$r= \left | \vec{v}\right | = \sqrt{3^2+5^2}=\sqrt{34}$ .

So, our vector written in polar form is

$\vec{v} = \left \langle \sqrt{34} \angle 59.04^\circ \right \rangle$

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Example Question #1 : Express A Vector In Polar Form

Express the vector $\left \langle 1,-1\right \rangle$ in polar form.

Possible Answers:

$\left(1, -\frac{\pi}{4}\right)$

$\left(\sqrt2, -\frac{3\pi}{4}\right)$

$\left(\sqrt2, \frac{3\pi}{4}\right)$

$\left(\sqrt2, -\frac{\pi}{4}\right)$

$\left(\frac{\sqrt2}{2}, \frac{3\pi}{4}\right)$

Correct answer:

$\left(\sqrt2, -\frac{\pi}{4}\right)$

Explanation:

To convert a point or a vector to its polar form, use the following equations to determine the magnitude and the direction.

$M=\sqrt{x^2+y^2}$

$\theta=tan^-^1 (\frac{y}{x})$

Substitute the vector $\left \langle 1,-1\right \rangle$ to the equations to find the magnitude and the direction.

$M=\sqrt{(1)^2+(-1)^2}= \sqrt2$

$\theta=tan^-^1 (\frac{-1}{1}) = tan^-^1 (-1) = -\frac{\pi}{4}$

The polar form is: $(M,\theta)= (\sqrt2, -\frac{\pi}{4})$

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Example Question #2 : Express A Vector In Polar Form

Express $\left \langle 3,6\right \rangle$ in polar form in degrees.

Possible Answers:

(18,1.107)

(9,63.434)

$(9,\frac{\pi}{2})$

$(\sqrt{45},63.435)$

$(\sqrt{45},1.107)$

Correct answer:

$(\sqrt{45},63.435)$

Explanation:

The polar form of the vector is: $(R,\theta)$

Find .

$R=\sqrt{ x^2+ y^2}=\sqrt{ 3^2+ 6^2}=\sqrt{9+36}=\sqrt{45}$

Find the angle.

$\theta= tan^-^1\left(\frac{y}{x}\right)= tan^-^1\left(\frac{6}{3}\right)=63.435$

$(R,\theta)=(\sqrt{45},63.435)$

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Precalculus : Matrices and Vectors

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Find A Direction Vector When Given Two Points

Example Question #1 : Find A Direction Vector When Given Two Points

Example Question #1 : Find A Direction Vector When Given Two Points

Example Question #101 : Matrices And Vectors

Example Question #12 : Find A Direction Vector When Given Two Points

Example Question #11 : Find A Direction Vector When Given Two Points

Example Question #12 : Find A Direction Vector When Given Two Points

Example Question #1 : Express A Vector In Polar Form

Example Question #1 : Express A Vector In Polar Form

Example Question #2 : Express A Vector In Polar Form

All Precalculus Resources

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