Exam 8: Polar Coordinates; Vectors

Find the indicated cross product. - $\mathbf { v } = \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k } , \quad \mathbf { w } = 5 \mathbf { i } - 5 \mathbf { j } - \mathbf { k }$ Find $\mathbf { w } \times \mathbf { v }$ .

Solve the problem. -If $\mathbf { v } = - 3 i + 2 j$ , find $\| v \|$ .

Find the indicated cross product. - $\mathbf { v } = - 6 \mathbf { i } + 2 \mathbf { j } , \mathbf { w } = 2 \mathbf { i } - 4 \mathbf { k }$ Find $\mathbf { w } \times \mathbf { v }$ .

State whether the vectors are parallel, orthogonal, or neither. - $\mathbf { v } = 4 \mathbf { i } - 2 \mathbf { j } , \quad \mathbf { w } = 4 \mathbf { i } + 2 \mathbf { j }$

Find the unit vector having the same direction as v. - $\mathbf { v } = - 4 \mathrm { j }$

Find all the complex roots. Leave your answers in polar form with the argument in degrees. -The complex fifth roots of $- 2 i$

Plot the point given in polar coordinates. - $\left(3, \frac{7 \pi}{6}\right)$

Choose the one alternative that best completes the statement or answers the question. The polar coordinates of a point are given. Find the rectangular coordinates of the point. - $\left( 4,70 ^ { \circ } \right)$ Round the rectangular coordinates to two decimal places.

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

Find the dot product v · w. - $\mathbf { v } = - \mathbf { i } - \mathbf { j } , \quad \mathbf { w } = \mathbf { i } + \mathbf { j }$

Choose the one alternative that best completes the statement or answers the question. Find the direction angles of the vector. Round to the nearest degree, if necessary. - $\mathbf { v } = 4 \mathbf { i } + 2 \mathbf { j } - 3 \mathbf { k }$

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

Find the position vector for the vector having initial point P and terminal point Q. - $\mathrm { P } = ( - 3,2 , - 3 )$ and $\mathrm { Q } = ( 2,0,1 )$

Choose the one alternative that best completes the statement or answers the question. Find the requested vector. - $\mathbf { v } = - 4 \mathbf { i } - 5 \mathbf { j } + \mathbf { k } , \quad \mathbf { w } = - 5 \mathbf { i } + 3 \mathbf { j } - \mathbf { k }$ Find a vector orthogonal to both $\mathbf { v }$ and $\mathbf { w }$ .

Find the value of the determinant. - $\left| \begin{array} { r r } - 6 & 9 \\- 7 & - 1\end{array} \right|$

Find the distance from P1 to P2. - $P _ { 1 } = ( 0 , - 4,4 ) \text { and } P _ { 2 } = ( - 1,1 , - 3 )$

Plot the point given in polar coordinates. - $\left( 2,360 ^ { 0 } \right)$

Match the graph to one of the polar equations. -

Identify and graph the polar equation. - $r^{2}=4 \cos (2 \theta)$

$36.86 \mathbf { i } + 25.81 \mathbf { j }$ -Two forces, $\mathbf { F } _ { 1 }$ of magnitude 60 newtons ( $( \mathrm { N } )$ and $\mathbf { F } _ { 2 }$ 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 $\mathbf { F } _ { \mathbf { 1 } } + \mathbf { F } _ { 2 }$ . Round the direction and magnitude to two decimal places.