Question 1

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

Accepted Answer

A) 
 
logarithmic spiral 
B) 
 
logarithmic spiral 
C) 
 
logarithmic spiral 
D) 
 
logarithmic spiral 
A) 
 
logarithmic spiral 
B) 
 
logarithmic spiral 
C) 
 
logarithmic spiral 
D) 
 
logarithmic spiral

Question 2

Solve the problem.
-If $\mathbf { v } = 3 \mathbf { i } + 4 \mathbf { j }$, find $\| \mathbf { v } \|$.

Accepted Answer

A)  5 
B)  7 
C)  $\sqrt { 5 }$ 
D)  25 
A)  5 
B)  7 
C)  $\sqrt { 5 }$ 
D)  25

Question 3

Find the dot product v · w.
-$\mathbf { v } = \mathrm { i } + \mathrm { j }$ and $\mathbf { w } = \mathrm { i } + \mathrm { j } - \mathbf { k }$

Accepted Answer

A)  1 
B)  3 
C)  $- 2$ 
D)  2 
A)  1 
B)  3 
C)  $- 2$ 
D)  2

Question 4

Write the vector v in the form ai + bj, given its magnitude v and the angle  it makes with the positive x-axis.
-$$\| \mathbf { v } \| = 7 , \quad \alpha = 60 ^ { \circ }$$

Accepted Answer

A)  $v = 7 \left( - \frac { 7 } { 2 } i - \frac { 7 \sqrt { 3 } } { 2 } j \right)$ 
B)  $v = 7 \left( \frac { 7 \sqrt { 3 } } { 2 } \mathrm { i } + \frac { 7 } { 2 } \mathrm { j } \right)$ 
C)  $v = 7 \left( \frac { \sqrt { 2 } } { 2 } \mathrm { i } + \frac { \sqrt { 2 } } { 2 } \mathrm { j } \right)$ 
D)  $v = 7 \left( \frac { 7 } { 2 } \mathrm { i } + \frac { 7 \sqrt { 3 } } { 2 } \mathrm { j } \right)$ 
A)  $v = 7 \left( - \frac { 7 } { 2 } i - \frac { 7 \sqrt { 3 } } { 2 } j \right)$ 
B)  $v = 7 \left( \frac { 7 \sqrt { 3 } } { 2 } \mathrm { i } + \frac { 7 } { 2 } \mathrm { j } \right)$ 
C)  $v = 7 \left( \frac { \sqrt { 2 } } { 2 } \mathrm { i } + \frac { \sqrt { 2 } } { 2 } \mathrm { j } \right)$ 
D)  $v = 7 \left( \frac { 7 } { 2 } \mathrm { i } + \frac { 7 \sqrt { 3 } } { 2 } \mathrm { j } \right)$

Question 5

Use the vectors in the figure below to graph the following vector.  
-$\mathbf { u } + \mathrm { z }$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 6

Match the graph to one of the polar equations.
-

Accepted Answer

A)  $r = 3 + \sin \theta$ 
B)  $r = 6 \sin \theta$ 
C)  $r = 3 + \cos \theta$ 
D)  $r = 6 \cos \theta$ 
A)  $r = 3 + \sin \theta$ 
B)  $r = 6 \sin \theta$ 
C)  $r = 3 + \cos \theta$ 
D)  $r = 6 \cos \theta$

Question 7

Find the position vector for the vector having initial point P and terminal point Q.
-$\mathrm { P } = ( 0,0,0 )$ and $\mathrm { Q } = ( - 3,3,4 )$

Accepted Answer

A)  $\mathbf { v } = - 3 \mathrm { i } + 3 \mathrm { j } + 4 \mathbf { k }$ 
B)  $v = - 3 i + 3 j - 4 k$ 
C)  $\mathbf { v } = 4 \mathbf { i } + 3 \mathbf { j } - 3 \mathbf { k }$ 
D)  $\mathbf { v } = 3 \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }$ 
A)  $\mathbf { v } = - 3 \mathrm { i } + 3 \mathrm { j } + 4 \mathbf { k }$ 
B)  $v = - 3 i + 3 j - 4 k$ 
C)  $\mathbf { v } = 4 \mathbf { i } + 3 \mathbf { j } - 3 \mathbf { k }$ 
D)  $\mathbf { v } = 3 \mathbf { i } - 3 \mathbf { j } - 4 \mathbf { k }$

Question 8

Find the unit vector having the same direction as v.
-$$\mathbf { v } = - 12 \mathbf { i } + 5 \mathbf { j }$$

Accepted Answer

A)  $\mathbf { u } = - \frac { 12 } { 13 } \mathbf { i } + \frac { 5 } { 13 } \mathbf { j }$ 
B)  $\mathbf { u } = - \frac { 13 } { 12 } \mathbf { i } + \frac { 13 } { 5 } \mathbf { j }$ 
C)  $\mathbf { u } = - \frac { 5 } { 13 } \mathbf { i } + \frac { 12 } { 13 } \mathbf { j }$ 
D)  $\mathbf { u } = - 156 \mathbf { i } + 65 \mathbf { j }$ 
A)  $\mathbf { u } = - \frac { 12 } { 13 } \mathbf { i } + \frac { 5 } { 13 } \mathbf { j }$ 
B)  $\mathbf { u } = - \frac { 13 } { 12 } \mathbf { i } + \frac { 13 } { 5 } \mathbf { j }$ 
C)  $\mathbf { u } = - \frac { 5 } { 13 } \mathbf { i } + \frac { 12 } { 13 } \mathbf { j }$ 
D)  $\mathbf { u } = - 156 \mathbf { i } + 65 \mathbf { j }$

Question 9

Write the vector v in the form ai + bj, given its magnitude v and the angle  it makes with the positive x-axis.
-$$\| v \| = 7 , \alpha = 270 ^ { \circ }$$

Accepted Answer

A)  $\mathbf { v } = - 7 \mathbf { i }$ 
B)  $\mathbf { v } = - 7 \mathrm { i } + 7 \mathrm { j } )$ 
C)  $\mathbf { v } = - 7 \mathbf { i } - 7 \mathbf { j }$ 
D)  $\mathbf { v } = - 7 \mathrm { j }$ 
A)  $\mathbf { v } = - 7 \mathbf { i }$ 
B)  $\mathbf { v } = - 7 \mathrm { i } + 7 \mathrm { j } )$ 
C)  $\mathbf { v } = - 7 \mathbf { i } - 7 \mathbf { j }$ 
D)  $\mathbf { v } = - 7 \mathrm { j }$

Question 10

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.
-$$\mathbf { v } = - 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }$$

Accepted Answer

A)  $\alpha = 119 ^ { \circ } , \beta = 43 ^ { \circ } , \gamma = 166 ^ { \circ }$ 
B)  $\alpha = 115 ^ { \circ } , \beta = 51 ^ { \circ } , \gamma = 147 ^ { \circ }$ 
C)  $\alpha = 112 ^ { \circ } , \beta = 124 ^ { \circ } , \gamma = 138 ^ { \circ }$ 
D)  $\alpha = 113 ^ { \circ } , \beta = 55 ^ { \circ } , \gamma = 140 ^ { \circ }$ 
A)  $\alpha = 119 ^ { \circ } , \beta = 43 ^ { \circ } , \gamma = 166 ^ { \circ }$ 
B)  $\alpha = 115 ^ { \circ } , \beta = 51 ^ { \circ } , \gamma = 147 ^ { \circ }$ 
C)  $\alpha = 112 ^ { \circ } , \beta = 124 ^ { \circ } , \gamma = 138 ^ { \circ }$ 
D)  $\alpha = 113 ^ { \circ } , \beta = 55 ^ { \circ } , \gamma = 140 ^ { \circ }$

Question 11

Use the given vectors to find the indicated expression.
-$$\mathbf { v } = - 4 \mathbf { i } - 3 \mathbf { j } + 2 \mathbf { k } , \quad \mathbf { w } = - 5 \mathbf { i } - 5 \mathbf { j } + 4 \mathbf { k } , \quad \mathbf { u } = 4 \mathbf { i } + 4 \mathbf { j } - 4 \mathbf { k }$$
Find $\mathbf { w } \cdot ( \mathbf { v } \times \mathbf { u } )$.

Accepted Answer

A)  148 
B)  4 
C)  172 
D)  0 
A)  148 
B)  4 
C)  172 
D)  0

Question 12

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

Accepted Answer

A)  $66 ^ { \circ }$ 
B)  $0 ^ { \circ }$ 
C)  $35.3 ^ { \circ }$ 
D)  $90 ^ { \circ }$ 
A)  $66 ^ { \circ }$ 
B)  $0 ^ { \circ }$ 
C)  $35.3 ^ { \circ }$ 
D)  $90 ^ { \circ }$

Question 13

Test the equation for symmetry with respect to the given axis, line, or pole.
-$r = 3 + 3 \cos \theta ;$ the line $\theta = \frac { \pi } { 2 }$

Accepted Answer

A)  Symmetric with respect to the line $\theta = \frac { \pi } { 2 }$ 
B)  May or may not be symmetric with respect to the line $\theta = \frac { \pi } { 2 }$ 
A)  Symmetric with respect to the line $\theta = \frac { \pi } { 2 }$ 
B)  May or may not be symmetric with respect to the line $\theta = \frac { \pi } { 2 }$

Question 14

Solve the problem. Leave your answer in polar form.
-$$\begin{array} { l } 
z = 2 + 2 i \
w = \sqrt { 3 } - i
\end{array}$$
Find zw.

Accepted Answer

A)  $4 \sqrt { 2 } \left( \cos \frac { 23 \pi } { 12 } + i \sin \frac { 23 \pi } { 12 } \right)$ 
B)  $4 \left( \cos \frac { \pi } { 12 } + i \sin \frac { \pi } { 12 } \right)$ 
C)  $4 \left( \cos \frac { 23 \pi } { 12 } + i \sin \frac { 23 \pi } { 12 } \right)$ 
D)  $4 \sqrt { 2 } \left( \cos \frac { \pi } { 12 } + i \sin \frac { \pi } { 12 } \right)$ 
A)  $4 \sqrt { 2 } \left( \cos \frac { 23 \pi } { 12 } + i \sin \frac { 23 \pi } { 12 } \right)$ 
B)  $4 \left( \cos \frac { \pi } { 12 } + i \sin \frac { \pi } { 12 } \right)$ 
C)  $4 \left( \cos \frac { 23 \pi } { 12 } + i \sin \frac { 23 \pi } { 12 } \right)$ 
D)  $4 \sqrt { 2 } \left( \cos \frac { \pi } { 12 } + i \sin \frac { \pi } { 12 } \right)$

Question 15

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w.
-$$\mathbf { v } = \mathrm { i } + 5 \mathrm { j } , \quad \mathbf { w } = \mathrm { i } + \mathrm { j }$$

Accepted Answer

A)  $\mathbf { v } _ { 1 } = \frac { 7 } { 2 } \mathrm { i } + \frac { 7 } { 2 } \mathrm { j } , \mathrm { v } _ { 2 } = - \frac { 5 } { 2 } \mathrm { i } + \frac { 3 } { 2 } \mathrm { j }$ 
B)  $\left. \mathbf { v } _ { 1 } = 3 \mathbf { i } + 3 \mathbf { j } \right) , \mathbf { v } _ { 2 } = - 2 \mathbf { i } + 2 \mathbf { j }$ 
C)  $\mathbf { v } _ { 1 } = 6 \mathbf { i } + 6 \mathbf { j } , \mathbf { v } _ { 2 } = - 4 \mathbf { i } + 4 \mathbf { j }$ 
D)  $\mathbf { v } _ { 1 } = 3 \mathbf { i } + 3 \mathbf { j } , \mathbf { v } _ { 2 } = 2 \mathbf { i } - 2 \mathbf { j }$ 
A)  $\mathbf { v } _ { 1 } = \frac { 7 } { 2 } \mathrm { i } + \frac { 7 } { 2 } \mathrm { j } , \mathrm { v } _ { 2 } = - \frac { 5 } { 2 } \mathrm { i } + \frac { 3 } { 2 } \mathrm { j }$ 
B)  $\left. \mathbf { v } _ { 1 } = 3 \mathbf { i } + 3 \mathbf { j } \right) , \mathbf { v } _ { 2 } = - 2 \mathbf { i } + 2 \mathbf { j }$ 
C)  $\mathbf { v } _ { 1 } = 6 \mathbf { i } + 6 \mathbf { j } , \mathbf { v } _ { 2 } = - 4 \mathbf { i } + 4 \mathbf { j }$ 
D)  $\mathbf { v } _ { 1 } = 3 \mathbf { i } + 3 \mathbf { j } , \mathbf { v } _ { 2 } = 2 \mathbf { i } - 2 \mathbf { j }$

Question 16

Use the given vectors to find the indicated expression.
-$$\mathbf { v } = 2 \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } , \quad \mathbf { w } = - 4 \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k } , \quad \mathbf { u } = 5 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k }$$
Find $\mathbf { u } \cdot ( \mathbf { v } \times \mathbf { w } )$.

Accepted Answer

A)  48 
B)  24 
C)  58 
D)  $- 24$ 
A)  48 
B)  24 
C)  58 
D)  $- 24$

Question 17

Graph the polar equation.
-$r=\frac{5}{1-5 \sin \theta}$

Accepted Answer

A) 
  
B) 
  
C) 
  
D) 
  
A) 
  
B) 
  
C) 
  
D)

Question 18

Write the vector v in the form ai + bj, given its magnitude v and the angle  it makes with the positive x-axis.
-$$\| \mathbf { v } \| = 14 , \quad \alpha = 0 ^ { \circ }$$

Accepted Answer

A)  $v = 14 \mathbf { i }$ 
B)  $v = 14 i + 14 j$ 
C)  $\mathbf { v } = 14 \mathbf { j }$ 
D)  $\mathbf { v } = - 14 \mathrm { j }$ 
A)  $v = 14 \mathbf { i }$ 
B)  $v = 14 i + 14 j$ 
C)  $\mathbf { v } = 14 \mathbf { j }$ 
D)  $\mathbf { v } = - 14 \mathrm { j }$

Question 19

Find all the complex roots. Leave your answers in polar form with the argument in degrees.
-The complex fifth roots of $\sqrt { 3 } + i$

Accepted Answer

A)  $\sqrt [ 5 ] { 2 } \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right) , \sqrt [ 5 ] { 2 } \left( \cos 102 ^ { \circ } + i \sin 102 ^ { \circ } \right) , \sqrt [ 5 ] { 2 } \left( \cos 174 ^ { \circ } + i \sin 174 ^ { \circ } \right) , \sqrt [ 5 ] { 2 } \left( \cos 246 ^ { \circ } + i \sin 246 ^ { \circ } \right)$, $\sqrt [ 5 ] { 2 } \left( \cos 318 ^ { \circ } + i \sin 318 ^ { \circ } \right)$ 
B)  $32 \left( \cos 6 ^ { \circ } + i \sin 6 ^ { \circ } \right) , 32 \left( \cos 78 ^ { \circ } + i \sin 78 ^ { \circ } \right) , 32 \left( \cos 150 ^ { \circ } + i \sin 150 ^ { \circ } \right) , 32 \left( \cos 222 ^ { \circ } + i \sin 222 ^ { \circ } \right)$, $32 \left( \cos 294 ^ { \circ } + \mathrm { i } \sin 294 ^ { \circ } \right)$ 
C)  $32 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right) , 32 \left( \cos 102 ^ { \circ } + i \sin 102 ^ { \circ } \right) , 32 \left( \cos 174 ^ { \circ } + i \sin 174 ^ { \circ } \right) , 32 \left( \cos 246 ^ { \circ } + i \sin 246 ^ { \circ } \right)$, $32 \left( \cos 318 ^ { \circ } + i \sin 318 ^ { \circ } \right)$ 
D)  $\sqrt [ 5 ] { 2 } \left( \cos 6 ^ { \circ } + i \sin 6 ^ { \circ } \right) , \sqrt [ 5 ] { 2 } \left( \cos 78 ^ { \circ } + i \sin 78 ^ { \circ } \right) , \sqrt [ 5 ] { 2 } \left( \cos 150 ^ { \circ } + i \sin 150 ^ { \circ } \right) , \sqrt [ 5 ] { 2 } \left( \cos 222 ^ { \circ } + i \sin 222 ^ { \circ } \right)$, $\sqrt [ 5 ] { 2 } \left( \cos 294 ^ { \circ } + i \sin 294 ^ { \circ } \right)$ 
A)  $\sqrt [ 5 ] { 2 } \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right) , \sqrt [ 5 ] { 2 } \left( \cos 102 ^ { \circ } + i \sin 102 ^ { \circ } \right) , \sqrt [ 5 ] { 2 } \left( \cos 174 ^ { \circ } + i \sin 174 ^ { \circ } \right) , \sqrt [ 5 ] { 2 } \left( \cos 246 ^ { \circ } + i \sin 246 ^ { \circ } \right)$, $\sqrt [ 5 ] { 2 } \left( \cos 318 ^ { \circ } + i \sin 318 ^ { \circ } \right)$ 
B)  $32 \left( \cos 6 ^ { \circ } + i \sin 6 ^ { \circ } \right) , 32 \left( \cos 78 ^ { \circ } + i \sin 78 ^ { \circ } \right) , 32 \left( \cos 150 ^ { \circ } + i \sin 150 ^ { \circ } \right) , 32 \left( \cos 222 ^ { \circ } + i \sin 222 ^ { \circ } \right)$, $32 \left( \cos 294 ^ { \circ } + \mathrm { i } \sin 294 ^ { \circ } \right)$ 
C)  $32 \left( \cos 30 ^ { \circ } + i \sin 30 ^ { \circ } \right) , 32 \left( \cos 102 ^ { \circ } + i \sin 102 ^ { \circ } \right) , 32 \left( \cos 174 ^ { \circ } + i \sin 174 ^ { \circ } \right) , 32 \left( \cos 246 ^ { \circ } + i \sin 246 ^ { \circ } \right)$, $32 \left( \cos 318 ^ { \circ } + i \sin 318 ^ { \circ } \right)$ 
D)  $\sqrt [ 5 ] { 2 } \left( \cos 6 ^ { \circ } + i \sin 6 ^ { \circ } \right) , \sqrt [ 5 ] { 2 } \left( \cos 78 ^ { \circ } + i \sin 78 ^ { \circ } \right) , \sqrt [ 5 ] { 2 } \left( \cos 150 ^ { \circ } + i \sin 150 ^ { \circ } \right) , \sqrt [ 5 ] { 2 } \left( \cos 222 ^ { \circ } + i \sin 222 ^ { \circ } \right)$, $\sqrt [ 5 ] { 2 } \left( \cos 294 ^ { \circ } + i \sin 294 ^ { \circ } \right)$

Question 20

Write the vector v in the form ai + bj, given its magnitude v and the angle  it makes with the positive x-axis.
-$$\| \mathbf { v } \| = 15 , \alpha = 45 ^ { \circ }$$

Accepted Answer

A)  $\mathbf { v } = 15 \left( \frac { 15 \sqrt { 3 } } { 2 } \mathbf { i } + \frac { 15 } { 2 } \mathbf { j } \right)$ 
B)  $\mathbf { v } = 15 \left( \frac { 15 } { 2 } \mathrm { i } + \frac { 15 \sqrt { 3 } } { 2 } \mathrm { j } \right)$ 
C)  $v = 15 \left( - \frac { \sqrt { 2 } } { 2 } i - \frac { \sqrt { 2 } } { 2 } j \right)$ 
D)  $\mathbf { v } = 15 \left( \frac { \sqrt { 2 } } { 2 } \mathrm { i } + \frac { \sqrt { 2 } } { 2 } \mathrm { j } \right)$ 
A)  $\mathbf { v } = 15 \left( \frac { 15 \sqrt { 3 } } { 2 } \mathbf { i } + \frac { 15 } { 2 } \mathbf { j } \right)$ 
B)  $\mathbf { v } = 15 \left( \frac { 15 } { 2 } \mathrm { i } + \frac { 15 \sqrt { 3 } } { 2 } \mathrm { j } \right)$ 
C)  $v = 15 \left( - \frac { \sqrt { 2 } } { 2 } i - \frac { \sqrt { 2 } } { 2 } j \right)$ 
D)  $\mathbf { v } = 15 \left( \frac { \sqrt { 2 } } { 2 } \mathrm { i } + \frac { \sqrt { 2 } } { 2 } \mathrm { j } \right)$

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

Solve the problem. -If $\mathbf { v } = 3 \mathbf { i } + 4 \mathbf { j }$ , find $\| \mathbf { v } \|$ .

Find the dot product v · w. - $\mathbf { v } = \mathrm { i } + \mathrm { j }$ and $\mathbf { w } = \mathrm { i } + \mathrm { j } - \mathbf { k }$

Write the vector v in the form ai + bj, given its magnitude v and the angle  it makes with the positive x-axis. - $\| \mathbf { v } \| = 7 , \quad \alpha = 60 ^ { \circ }$

Match the graph to one of the polar equations. -

Find the position vector for the vector having initial point P and terminal point Q. - $\mathrm { P } = ( 0,0,0 )$ and $\mathrm { Q } = ( - 3,3,4 )$

Find the unit vector having the same direction as v. - $\mathbf { v } = - 12 \mathbf { i } + 5 \mathbf { j }$

Write the vector v in the form ai + bj, given its magnitude v and the angle  it makes with the positive x-axis. - $\| v \| = 7 , \alpha = 270 ^ { \circ }$

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. - $\mathbf { v } = - 2 \mathbf { i } + 3 \mathbf { j } - 4 \mathbf { k }$

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

Test the equation for symmetry with respect to the given axis, line, or pole. - $r = 3 + 3 \cos \theta ;$ the line $\theta = \frac { \pi } { 2 }$

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

Decompose v into two vectors v1 and v2, where v1 is parallel to w and v2 is orthogonal to w. - $\mathbf { v } = \mathrm { i } + 5 \mathrm { j } , \quad \mathbf { w } = \mathrm { i } + \mathrm { j }$

Graph the polar equation. - $r=\frac{5}{1-5 \sin \theta}$ $Graph the polar equation. - r=\frac{5}{1-5 \sin \theta}$

Write the vector v in the form ai + bj, given its magnitude v and the angle  it makes with the positive x-axis. - $\| \mathbf { v } \| = 14 , \quad \alpha = 0 ^ { \circ }$

Find all the complex roots. Leave your answers in polar form with the argument in degrees. -The complex fifth roots of $\sqrt { 3 } + i$

Write the vector v in the form ai + bj, given its magnitude v and the angle  it makes with the positive x-axis. - $\| \mathbf { v } \| = 15 , \alpha = 45 ^ { \circ }$

Functions and Their Graphs

Linear and Quadratic Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions

Analytic Trigonometry

Applications of Trigonometric Functions

Analytic Geometry

Systems of Equations and Inequalities

Sequences; Induction; the Binomial Theorem

Counting and Probability

A Preview of Calculus: the Limit, Derivative, and Integral of a Function

Foundations: a Prelude to Functions

Graphing Utilities

Filters

Exam 8: Polar Coordinates; Vectors

Identify and graph the polar equation. - r=4θr=4 \thetar=4θ

Solve the problem. -If v=3i+4j\mathbf { v } = 3 \mathbf { i } + 4 \mathbf { j }v=3i+4j , find ∥v∥\| \mathbf { v } \|∥v∥ .

Find the dot product v · w. - v=i+j\mathbf { v } = \mathrm { i } + \mathrm { j }v=i+j and w=i+j−k\mathbf { w } = \mathrm { i } + \mathrm { j } - \mathbf { k }w=i+j−k

Write the vector v in the form ai + bj, given its magnitude v and the angle  it makes with the positive x-axis. - ∥v∥=7,α=60∘\| \mathbf { v } \| = 7 , \quad \alpha = 60 ^ { \circ }∥v∥=7,α=60∘

Use the vectors in the figure below to graph the following vector. - u+z\mathbf { u } + \mathrm { z }u+z