10.8 Vectors

Find the scalar multiple 3 $u$ given $u$ $= \langle 5 , 4 \rangle .$

Given $u$ $= \langle 3 , - 2 \rangle$ and $v$ $= \langle - 1 , 4 \rangle ,$ find a new vector w = 3u + 2v.

Given a vector $v$ with initial point $P = \left(2 , −6\right)$ and terminal point $Q = \left(−6 , 6\right) ,$ write the vector in terms of $i$ and $j .$

Given initial point $P_{1} = \left(- 1 , 3\right)$ and terminal point $P_{2} = \left(2 , 7\right) ,$ write the vector $v$ in terms of $i$ and $j .$

Write the vector $u$ with initial point $P = \left(- 1 , 6\right)$ and terminal point $Q = \left(7 , - 5\right)$ in terms of $i$ and $j .$

Find the sum of $v_{1} = 2 i - 3 j$ and $v_{2} = 4 i + 5 j .$

Given a vector with length 7 and an angle of 135°, write it in component form.

A vector travels from the origin to the point $\left(3 , 5\right) .$ Write the vector in terms of magnitude and direction.

Find the dot product of $v = \langle 5 , 12 \rangle$ and $u = \langle −3 , 4 \rangle .$

What are the characteristics of the letters that are commonly used to represent vectors?

What are $i$ and $j ,$ and what do they represent?

When a unit vector is expressed as $\langle a , b \rangle ,$ which letter is the coefficient of the $i$ and which the $j ?$

Given a vector with initial point $\left(- 4 , 2\right)$ and terminal point $\left(3 , - 3\right) ,$ find an equivalent vector whose initial point is $\left(0 , 0\right) .$ Write the vector in component form $\langle a , b \rangle .$

Determine whether the two vectors $u$ and $v$ are equal, where $u$ has an initial point $P_{1}$ and a terminal point $P_{2}$ and $v$ has an initial point $P_{3}$ and a terminal point $P_{4}$.

$P_{1} = \left(5 , 1\right) , P_{2} = \left(3 , - 2\right) , P_{3} = \left(- 1 , 3\right) ,$ and $P_{4} = \left(9 , - 4\right)$

Determine whether the two vectors $u$ and $v$ are equal, where $u$ has an initial point $P_{1}$ and a terminal point $P_{2}$ and $v$ has an initial point $P_{3}$ and a terminal point $P_{4}$.

$P_{1} = \left(- 1 , - 1\right) , P_{2} = \left(- 4 , 5\right) , P_{3} = \left(- 10 , 6\right) ,$ and $P_{4} = \left(- 13 , 12\right)$

Determine whether the two vectors $u$ and $v$ are equal, where $u$ has an initial point $P_{1}$ and a terminal point $P_{2}$ and $v$ has an initial point $P_{3}$ and a terminal point $P_{4}$.

$P_{1} = \left(8 , 3\right) , P_{2} = \left(6 , 5\right) , P_{3} = \left(11 , 8\right) ,$ and $P_{4} = \left(9 , 10\right)$

Determine whether the two vectors $u$ and $v$ are equal, where $u$ has an initial point $P_{1}$ and a terminal point $P_{2}$ and $v$ has an initial point $P_{3}$ and a terminal point $P_{4}$.

Given initial point $P_{1} = \left(6 , 0\right)$ and terminal point $P_{2} = \left(- 1 , - 3\right) ,$ write the vector $v$ in terms of $i$ and $j .$

Use the vectors u = i + 5j, v = −2i− 3j,  and w = 4i − j.

Find 4v + 2u

Use the given vectors to compute u + v, u − v, and 2u − 3v.

$u = \langle - 3 , 4 \rangle , v = \langle - 2 , 1 \rangle$

Use the given vectors to compute u + v, u − v, and 2u − 3v.

Let v = 5i + 2j. Find a vector that is twice the length and points in the opposite direction as $v .$

Find a unit vector in the same direction as the given vector.

b = −2i + 5j

Find a unit vector in the same direction as the given vector.

$d = - \frac{1}{3} i + \frac{5}{2} j$

Find a unit vector in the same direction as the given vector.

u = −14i + 2j

Find the magnitude and direction of the vector, $0 \leq \theta < 2 \pi .$

$\langle 6 , 5 \rangle$

Find the magnitude and direction of the vector, $0 \leq \theta < 2 \pi .$

$\langle −4 , −6 \rangle$

Find the magnitude and direction of the vector, $0 \leq \theta < 2 \pi .$

Given u = −i − j and v = i + 5j, calculate $u \cdot v .$

Find the magnitude and direction of the vector, $0 \leq \theta < 2 \pi .$

Given u $= \langle - 1 , 6 \rangle$ and v $= \langle 6 , - 1 \rangle ,$ calculate $u \cdot v .$

Use the given magnitude and direction in standard position, write the vector in component form.

$\left|v\right| = 6 , \theta = 45 \circ$

Use the given magnitude and direction in standard position, write the vector in component form.

$\left|v\right| = 2 , \theta = 300°$

Use the given magnitude and direction in standard position, write the vector in component form.

A 60-pound box is resting on a ramp that is inclined 12°. Rounding to the nearest tenth,

1. ⓐ Find the magnitude of the normal (perpendicular) component of the force.
2. ⓑ Find the magnitude of the component of the force that is parallel to the ramp.

Use the given magnitude and direction in standard position, write the vector in component form.

Find the magnitude of the horizontal and vertical components of a vector with magnitude 8 pounds pointed in a direction of 27° above the horizontal. Round to the nearest hundredth.

Use the given magnitude and direction in standard position, write the vector in component form.

Find the magnitude of the horizontal and vertical components of a vector with magnitude 5 pounds pointed in a direction of 55° above the horizontal. Round to the nearest hundredth.

A woman leaves home and walks 3 miles west, then 2 miles southwest. How far from home is she, and in what direction must she walk to head directly home?

A man starts walking from home and walks 4 miles east, 2 miles southeast, 5 miles south, 4 miles southwest, and 2 miles east. How far has he walked? If he walked straight home, how far would he have to walk?

A man starts walking from home and walks 3 miles at 20° north of west, then 5 miles at 10° west of south, then 4 miles at 15° north of east. If he walked straight home, how far would he have to the walk, and in what direction?

An airplane is heading north at an airspeed of 600 km/hr, but there is a wind blowing from the southwest at 80 km/hr. How many degrees off course will the plane end up flying, and what is the plane’s speed relative to the ground?

An airplane needs to head due north, but there is a wind blowing from the southwest at 60 km/hr. The plane flies with an airspeed of 550 km/hr. To end up flying due north, how many degrees west of north will the pilot need to fly the plane?

As part of a video game, the point $\left(5 , 7\right)$ is rotated counterclockwise about the origin through an angle of 35°. Find the new coordinates of this point.

Two children are throwing a ball back and forth straight across the back seat of a car. The ball is being thrown 10 mph relative to the car, and the car is traveling 25 mph down the road. If one child doesn't catch the ball, and it flies out the window, in what direction does the ball fly (ignoring wind resistance)?

A 50-pound object rests on a ramp that is inclined 19°. Find the magnitude of the components of the force parallel to and perpendicular to (normal) the ramp to the nearest tenth of a pound.

Suppose a body has a force of 10 pounds acting on it to the right, 25 pounds acting on it ─135° from the horizontal, and 5 pounds acting on it directed 150° from the horizontal. What single force is the resultant force acting on the body?

Suppose a body has a force of 3 pounds acting on it to the left, 4 pounds acting on it upward, and 2 pounds acting on it 30° from the horizontal. What single force is needed to produce a state of equilibrium on the body? Draw the vector.

Assume $\alpha$ is opposite side $a , \beta$ is opposite side $b ,$ and $\gamma$ is opposite side $c .$ Solve each triangle, if possible. Round each answer to the nearest tenth.

$\beta = 50° , a = 105 , b = 45$

Convert $\left(- 2 , \frac{3 \pi}{2}\right)$ to rectangular coordinates.

Convert $\left(- 9 , - 4\right)$ to polar coordinates.

Convert the given Cartesian equation to a polar equation.

$x^{2} + y^{2} = 64$

Convert the given polar equation to a Cartesian equation.

$r = 7 \text{cos} \theta$

Test each equation for symmetry.

$r = 4 + 4 \sin \theta$

Find the absolute value of each complex number.

$4 - ​ 3 i$

Write the complex number in polar form.

$\frac{1}{2} - \frac{\sqrt{3}}{2} ​ i$

Convert the complex number from polar to rectangular form.

$z = 3 cis \left(40°\right)$

Find the product $z_{1} z_{2}$ in polar form.

$z_{1} = 10 cis \left(\frac{\pi}{6}\right)$

$z_{2} = 6 cis \left(\frac{\pi}{3}\right)$

Find the quotient $\frac{z_{1}}{z_{2}}$ in polar form.

$z_{1} = 27 cis \left(\frac{5 \pi}{3}\right)$

$z_{2} = 9 cis \left(\frac{\pi}{3}\right)$

Find the powers of each complex number in polar form.

Find $z^{2}$ when $z = 5 cis \left(\frac{3 \pi}{4}\right)$

Evaluate each root.

Evaluate the square root of $z$ when $z = 25 cis \left(\frac{3 \pi}{2}\right) .$

Eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

$\begin{cases} x \left(t\right) = - \cos t \\ y \left(t\right) = 2 \sin^{2} t \end{cases}$

Eliminate the parameter $t$ to rewrite the parametric equation as a Cartesian equation.

Parameterize the line from $\left(- 2 , 3\right)$ to $\left(4 , 7\right)$ so that the line is at $\left(- 2 , 3\right)$ at $t = 0$ and $\left(4 , 7\right)$ at $t = 1.$

Make a table of values for each set of parametric equations, graph the equations, and include an orientation; then write the Cartesian equation.

A ball is launched with an initial velocity of 80 feet per second at an angle of 40° to the horizontal. The ball is released at a height of 4 feet above the ground.

1. ⓐ Find the parametric equations to model the path of the ball.
2. ⓑ Where is the ball after 3…

Determine whether the two vectors, $u$ and $v ,$ are equal, where $u$ has an initial point $P_{1}$ and a terminal point $P_{2} ,$ and $v$ has an initial point $P_{3}$ and a terminal point $P_{4} .$

$P_{1} = \left(6 , 11\right) , P_{2} = \left(- 2 , 8\right) , P_{3} = \left(0 , - 1\right)$ and $P_{4} = \left(- 8 , 2\right)$

Use the vectors $u = 2 i - j , v = 4 i - 3 j ,$ and $w = - 2 i + 5 j$ to evaluate the expression.

2v − u + w

Find a unit vector in the same direction as the given vector.

b = −3i − j

Find the magnitude and direction of the vector.

$\langle −3 , −3 \rangle$

Calculate $u \cdot v .$

u = i + 4j and v = 4i + 3j

Assume $\alpha$ is opposite side $a , \beta$ is opposite side $b ,$ and $\gamma$ is opposite side $c .$ Solve the triangle, if possible, and round each answer to the nearest tenth, given $\beta = 68° , b = 21 , c = 16.$

A pilot flies in a straight path for 2 hours. He then makes a course correction, heading 15° to the right of his original course, and flies 1 hour in the new direction. If he maintains a constant speed of 575 miles per hour, how far is he from his starting position?

Convert $\left(2 , \frac{\pi}{3}\right)$ to rectangular coordinates.

Find the absolute value of the complex number $5 - 9 i .$

Convert the complex number from polar to rectangular form: $z = 5 \text{cis} \left(\frac{2 \pi}{3}\right) .$

Given $z_{1} = 8 cis \left(36°\right)$ and $z_{2} = 2 cis \left(15°\right) ,$ evaluate each expression.

$\frac{z_{1}}{z_{2}}$

Given $z_{1} = 8 cis \left(36°\right)$ and $z_{2} = 2 cis \left(15°\right) ,$ evaluate each expression.

$\sqrt{z_{1}}$

Given $z_{1} = 8 cis \left(36°\right)$ and $z_{2} = 2 cis \left(15°\right) ,$ evaluate each expression.

Eliminate the parameter $t$ to rewrite the following parametric equations as a Cartesian equation: $\left\{\right. \begin{matrix}x \left(t\right) = t + 1 \\ y \left(t\right) = 2 t^{2}\end{matrix} .$

Use the vectors u = i − 3j and v = 2i + 3j.

Find 2u − 3v.

Use the vectors u = i − 3j and v = 2i + 3j.

Find a unit vector in the same direction as $v .$