Exam 8: Polar Coordinates; Vectors

State whether the vectors are parallel, orthogonal, or neither. - $\mathbf { v } = - 2 \mathrm { i } - 2 \mathrm { j } , \quad \mathrm { w } = 3 \mathrm { i } + \mathrm { j }$

Describe the set of points (x, y, z) defined by the equation. -z = 0

State whether the vectors are parallel, orthogonal, or neither. -v = 4i + 2j, w = 8i + 4j

Find the indicated cross product. -v = -6i + 2j, w = 2i - 4k Find w × v.

The polar coordinates of a point are given. Find the rectangular coordinates of the point. - $\left( - 3 , - 135 ^ { \circ } \right)$

Use the given vectors to find the indicated expression. -v = -4i + 2j - 3k, u = -4i - 3j + 4k Find v × (-5u).

Write the expression in the standard form a + bi. - $\left[ \sqrt { 2 } \left( \cos \frac { 3 \pi } { 4 } + i \sin \frac { 3 \pi } { 4 } \right) \right] ^ { 4 }$

Use the given vectors to find the indicated expression. -v = 2i + 2j + 2k, w = -4i - 3j + 4k, u = 5i + 2j + 5k Find u · (v × w).

Solve the problem. Leave your answer in polar form. - z=10 4+i4 w=5 1+i1 Find $\frac { z } { w }$ .

Solve the problem. Leave your answer in polar form. - z=2+2i w=-i Find $z W$ .

Solve the problem. -Find a vector v whose magnitude is 26 and whose component in the i direction is twice the component in the j direction. A) $v = \frac { 52 } { 5 } \sqrt { 5 } i + \frac { 26 } { 5 } \sqrt { 5 } j \quad$ or $\quad \mathbf { v } = - \frac { 52 } { 5 } \sqrt { 5 } i - \frac { 26 } { 5 } \sqrt { 5 } \mathbf { j }$ B) $v = \frac { 11 } { 10 } \sqrt { 10 } i - \frac { 33 } { 10 } \sqrt { 10 } j \quad$ or $\quad v = - \frac { 11 } { 10 } \sqrt { 10 } i + \frac { 33 } { 10 } \sqrt { 10 } \mathbf { j }$ C) $v = - \frac { 33 } { 10 } \sqrt { 10 } \mathrm { i } + \frac { 11 } { 10 } \sqrt { 10 } \mathrm { j } \quad$ or $\quad \mathbf { v } = \frac { 33 } { 10 } \sqrt { 10 } \mathrm { i } - \frac { 11 } { 10 } \sqrt { 10 } \mathrm { j }$ D) $v = \frac { 11 } { 10 } \sqrt { 10 } \mathrm { i } + \frac { 33 } { 10 } \sqrt { 10 } \mathrm { j } \quad$ or $\quad \mathbf { v } = - \frac { 11 } { 10 } \sqrt { 10 } \mathrm { i } - \frac { 33 } { 10 } \sqrt { 10 } \mathbf { j }$

The letters x and y represent rectangular coordinates. Write the equation using polar coordinates (r, θ). - $y ^ { 2 } = 16 x$

Match the graph to one of the polar equations. - A) $r = 3 + \sin \theta$ B) $r = 3 + \cos \theta$ C) $r = 6 \cos \theta$ D) $r = 6 \sin \theta$

Solve the problem. -Which of the following vectors is orthogonal to 20i - 8j?

Write the complex number in rectangular form. - $8 \left( \cos \frac { \pi } { 6 } + i \sin \frac { \pi } { 6 } \right)$

Use the given vectors to find the indicated expression. -v = -3i + 5j - 4k, w = 2i + 4j - 2k, u = -4i - 4j - 3k Find v · (v × w).

Write the complex number in polar form. Express the argument in degrees, rounded to the nearest tenth, if necessary. - $- 5 i$

Solve the problem. -Two forces, $\mathrm { F } _ { 1 }$ of magnitude 60 newtons $( \mathrm { N } )$ and $\mathrm { F } _ { 2 }$ of magnitude 70 newtons, act on an object at angles of $40 ^ { \circ }$ and $130 ^ { \circ }$ (respectively) with the positive $x$ -axis. Find the direction and magnitude of the resultant force; that is, find $\mathbf { F } _ { \mathbf { 1 } } + \mathbf { F } _ { \mathbf { 2 } }$ . Round the direction and magnitude to two decimal places.

State whether the vectors are parallel, orthogonal, or neither. - $\mathbf { v } = - 3 \mathbf { i } - 5 \mathbf { j } , \quad \mathbf { w } = 2 \mathbf { i } - \mathbf { j }$

Solve the problem. -Find a unit vector normal to the plane containing u = -i + j + 4k and v = 2i - 3j + k.