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Identify and graph the polar equation. - r=4θr=4 \theta Identify and graph the polar equation. - r=4 \theta A) logarithmic spiral B) logarithmic spiral C) logarithmic spiral D) logarithmic spiral


A)
Identify and graph the polar equation. - r=4 \theta A) logarithmic spiral B) logarithmic spiral C) logarithmic spiral D) logarithmic spiral
logarithmic spiral

B)
Identify and graph the polar equation. - r=4 \theta A) logarithmic spiral B) logarithmic spiral C) logarithmic spiral D) logarithmic spiral
logarithmic spiral
C)
Identify and graph the polar equation. - r=4 \theta A) logarithmic spiral B) logarithmic spiral C) logarithmic spiral D) logarithmic spiral
logarithmic spiral

D)
Identify and graph the polar equation. - r=4 \theta A) logarithmic spiral B) logarithmic spiral C) logarithmic spiral D) logarithmic spiral
logarithmic spiral

E) A) and B)
F) A) and C)

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


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

E) C) and D)
F) B) and C)

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


A) 48
B) 24
C) 58
D) −24- 24

E) A) and B)
F) A) and C)

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Test the equation for symmetry with respect to the given axis, line, or pole. - r=3+3cos⁡θ;r = 3 + 3 \cos \theta ; the line θ=π2\theta = \frac { \pi } { 2 }


A) Symmetric with respect to the line θ=π2\theta = \frac { \pi } { 2 }
B) May or may not be symmetric with respect to the line θ=π2\theta = \frac { \pi } { 2 }

C) A) and B)
D) undefined

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


A) u=−1213i+513j\mathbf { u } = - \frac { 12 } { 13 } \mathbf { i } + \frac { 5 } { 13 } \mathbf { j }
B) u=−1312i+135j\mathbf { u } = - \frac { 13 } { 12 } \mathbf { i } + \frac { 13 } { 5 } \mathbf { j }
C) u=−513i+1213j\mathbf { u } = - \frac { 5 } { 13 } \mathbf { i } + \frac { 12 } { 13 } \mathbf { j }
D) u=−156i+65j\mathbf { u } = - 156 \mathbf { i } + 65 \mathbf { j }

E) B) and D)
F) B) and C)

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


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

E) None of the above
F) All of the above

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


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

E) B) and C)
F) A) and C)

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


A) v=15(1532i+152j) \mathbf { v } = 15 \left( \frac { 15 \sqrt { 3 } } { 2 } \mathbf { i } + \frac { 15 } { 2 } \mathbf { j } \right)
B) v=15(152i+1532j) \mathbf { v } = 15 \left( \frac { 15 } { 2 } \mathrm { i } + \frac { 15 \sqrt { 3 } } { 2 } \mathrm { j } \right)
C) v=15(−22i−22j) v = 15 \left( - \frac { \sqrt { 2 } } { 2 } i - \frac { \sqrt { 2 } } { 2 } j \right)
D) v=15(22i+22j) \mathbf { v } = 15 \left( \frac { \sqrt { 2 } } { 2 } \mathrm { i } + \frac { \sqrt { 2 } } { 2 } \mathrm { j } \right)

E) All of the above
F) A) and B)

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Graph the polar equation. - r=51−5sin⁡θr=\frac{5}{1-5 \sin \theta} Graph the polar equation. - r=\frac{5}{1-5 \sin \theta} A) B) C) D)


A)
Graph the polar equation. - r=\frac{5}{1-5 \sin \theta} A) B) C) D)

B)
Graph the polar equation. - r=\frac{5}{1-5 \sin \theta} A) B) C) D)
C)
Graph the polar equation. - r=\frac{5}{1-5 \sin \theta} A) B) C) D)

D)
Graph the polar equation. - r=\frac{5}{1-5 \sin \theta} A) B) C) D)

E) B) and C)
F) None of the above

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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=270∘\| v \| = 7 , \alpha = 270 ^ { \circ }


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

E) A) and B)
F) B) and D)

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


A) 25(cos⁡30∘+isin⁡30∘) ,25(cos⁡102∘+isin⁡102∘) ,25(cos⁡174∘+isin⁡174∘) ,25(cos⁡246∘+isin⁡246∘) \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) , 25(cos⁡318∘+isin⁡318∘) \sqrt [ 5 ] { 2 } \left( \cos 318 ^ { \circ } + i \sin 318 ^ { \circ } \right)
B) 32(cos⁡6∘+isin⁡6∘) ,32(cos⁡78∘+isin⁡78∘) ,32(cos⁡150∘+isin⁡150∘) ,32(cos⁡222∘+isin⁡222∘) 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(cos⁡294∘+isin⁡294∘) 32 \left( \cos 294 ^ { \circ } + \mathrm { i } \sin 294 ^ { \circ } \right)
C) 32(cos⁡30∘+isin⁡30∘) ,32(cos⁡102∘+isin⁡102∘) ,32(cos⁡174∘+isin⁡174∘) ,32(cos⁡246∘+isin⁡246∘) 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(cos⁡318∘+isin⁡318∘) 32 \left( \cos 318 ^ { \circ } + i \sin 318 ^ { \circ } \right)

D) 25(cos⁡6∘+isin⁡6∘) ,25(cos⁡78∘+isin⁡78∘) ,25(cos⁡150∘+isin⁡150∘) ,25(cos⁡222∘+isin⁡222∘) \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) , 25(cos⁡294∘+isin⁡294∘) \sqrt [ 5 ] { 2 } \left( \cos 294 ^ { \circ } + i \sin 294 ^ { \circ } \right)

E) C) and D)
F) None of the above

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


A) v1=72i+72j,v2=−52i+32j\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) v1=3i+3j) ,v2=−2i+2j\left. \mathbf { v } _ { 1 } = 3 \mathbf { i } + 3 \mathbf { j } \right) , \mathbf { v } _ { 2 } = - 2 \mathbf { i } + 2 \mathbf { j }
C) v1=6i+6j,v2=−4i+4j\mathbf { v } _ { 1 } = 6 \mathbf { i } + 6 \mathbf { j } , \mathbf { v } _ { 2 } = - 4 \mathbf { i } + 4 \mathbf { j }
D) v1=3i+3j,v2=2i−2j\mathbf { v } _ { 1 } = 3 \mathbf { i } + 3 \mathbf { j } , \mathbf { v } _ { 2 } = 2 \mathbf { i } - 2 \mathbf { j }

E) A) and D)
F) B) and C)

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


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

E) A) and B)
F) A) and C)

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


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

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

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

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

E) B) and C)
F) C) and D)

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


A) 1
B) 3
C) −2- 2
D) 2

E) A) and B)
F) None of the above

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


A) 42(cos⁡23π12+isin⁡23π12) 4 \sqrt { 2 } \left( \cos \frac { 23 \pi } { 12 } + i \sin \frac { 23 \pi } { 12 } \right)
B) 4(cos⁡π12+isin⁡π12) 4 \left( \cos \frac { \pi } { 12 } + i \sin \frac { \pi } { 12 } \right)
C) 4(cos⁡23π12+isin⁡23π12) 4 \left( \cos \frac { 23 \pi } { 12 } + i \sin \frac { 23 \pi } { 12 } \right)
D) 42(cos⁡π12+isin⁡π12) 4 \sqrt { 2 } \left( \cos \frac { \pi } { 12 } + i \sin \frac { \pi } { 12 } \right)

E) A) and C)
F) B) and C)

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


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

E) A) and B)
F) A) and C)

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


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

E) None of the above
F) B) and D)

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


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

E) B) and D)
F) B) and C)

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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 }


A) v=7(−72i−732j) v = 7 \left( - \frac { 7 } { 2 } i - \frac { 7 \sqrt { 3 } } { 2 } j \right)
B) v=7(732i+72j) v = 7 \left( \frac { 7 \sqrt { 3 } } { 2 } \mathrm { i } + \frac { 7 } { 2 } \mathrm { j } \right)
C) v=7(22i+22j) v = 7 \left( \frac { \sqrt { 2 } } { 2 } \mathrm { i } + \frac { \sqrt { 2 } } { 2 } \mathrm { j } \right)
D) v=7(72i+732j) v = 7 \left( \frac { 7 } { 2 } \mathrm { i } + \frac { 7 \sqrt { 3 } } { 2 } \mathrm { j } \right)

E) A) and D)
F) B) and C)

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