Exam 8: Polar Coordinates; Vectors

Find the angle between v and w. Round to one decimal place, if necessary. - $\mathbf { v } = 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } \text { and } \mathbf { w } = 4 \mathbf { i } + 5 \mathbf { j } + 2 \mathbf { k }$

Use the given vectors to find the indicated expression. - $v = - 3 i - 4 j + 4 k , \quad w = 2 i + 2 j - 3 k$ Find $v \times ( 2 w )$ .

Plot the complex number in the complex plane. - $- 6 i$

Transform the polar equation to an equation in rectangular coordinates. Then identify and graph the equation. - $\theta=\frac{\pi}{3}$

Plot the point given in polar coordinates. - $\left(-4,405^{\circ}\right)$

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w. - $\mathbf { v } = - 2 \mathbf { i } - 2 \mathbf { j } , \quad \mathbf { w } = 3 \mathbf { i } + \mathbf { j }$

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. - $\left( 400,130 ^ { \circ } \right)$ Round the rectangular coordinates to two decimal places.

Find the dot product v · w. - $v = - 4 i , \quad w = j$

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

Plot the point given in polar coordinates. - $\left( 4,30 ^ { \circ } \right)$

Identify and graph the polar equation. - $r=4 \sin (2 \theta)$

Identify and graph the polar equation. - $r=3-4 \sin \theta$

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. - $\mathbf { v } = 3 \mathbf { i } + 2 \mathbf { j } - 6 \mathbf { k }$

Plot the complex number in the complex plane. - $- 6 + i$

The rectangular coordinates of a point are given. Find polar coordinates for the point. - $( - 7,0 )$

Use the vectors in the figure below to graph the following vector. - $\mathbf { v } - \mathbf { w }$

Graph the polar equation. - $r = \frac { 4 } { 1 - \sin \theta }$

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.

Solve the problem. -If $\mathbf { v } = 5 \mathbf { i } + \mathbf { j }$ and $\mathbf { w } = 9 \mathrm { i } + \mathrm { j }$ , find $\| \mathbf { v } + \mathbf { w } \|$ .

Write the word or phrase that best completes each statement or answers the question. Solve the problem. -An airplane has an air speed of 550 miles per hour bearing N30°W. The wind velocity is 50 kilometers per hour in the direction N30°E. Find the resultant vector (with exact components) representing the path of the plane relative to the ground. To the nearest tenth, what is the ground speed of the plane? What is its direction?