Question 1

Find the area of the parallelogram that has the vectors as adjacent sides.  $\begin{array} { l } 
\mathbf { u } = \mathbf { i } + 3 \mathbf { j } + 3 \mathbf { k } \
\mathbf { v } = \mathbf { i } + \mathbf { k }
\end{array}$

Accepted Answer

A)  $\text { Area } = \sqrt { 22 } \text { square units }$ 
B)  $\text { Area } = \sqrt { 13 } \text { square units }$ 
C)  $\text { Area } = \sqrt { 17 } \text { square units }$ 
D)  $\text { Area } = \sqrt { 14 } \text { square units }$ 
E)  $\text { Area } = \sqrt { 23 } \text { square units }$ 
A)  $\text { Area } = \sqrt { 22 } \text { square units }$ 
B)  $\text { Area } = \sqrt { 13 } \text { square units }$ 
C)  $\text { Area } = \sqrt { 17 } \text { square units }$ 
D)  $\text { Area } = \sqrt { 14 } \text { square units }$ 
E)  $\text { Area } = \sqrt { 23 } \text { square units }$

Question 2

Find u × v and show that it is orthogonal to both u and v. $\mathbf { u } = ( 10,0,3 ) \quad \mathbf { v } = ( 10,0,0 )$

Accepted Answer

A)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = ( 0,30,0 ) \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$ 
B)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = ( 0 , - 30,0 ) \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$ 
C)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = ( 0,30,0 ) \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } \neq 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } \neq 0
\end{array}$ 
D)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = ( 30,0,0 ) \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$ 
E)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = ( 0,0,30 ) \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } \neq 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } \neq 0
\end{array}$ 
A)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = ( 0,30,0 ) \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$ 
B)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = ( 0 , - 30,0 ) \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$ 
C)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = ( 0,30,0 ) \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } \neq 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } \neq 0
\end{array}$ 
D)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = ( 30,0,0 ) \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } = 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } = 0
\end{array}$ 
E)  $\begin{array} { l } 
\mathbf { u } \times \mathbf { v } = ( 0,0,30 ) \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { u } \neq 0 \
( \mathbf { u } \times \mathbf { v } ) \cdot \mathbf { v } \neq 0
\end{array}$

Question 3

Find a unit vector in the direction of u. u = $( 11 , - 13,7 )$

Accepted Answer

A)  $\frac { \sqrt { 31 } } { \sqrt { 339 } } ( 11 , - 13,7 )$ 
B)  $\frac { 1 } { \sqrt { 339 } } ( 11 , - 13,7 )$ 
C)  $( 11 , - 13,7 )$ 
D)  $\sqrt { 339 } ( 11 , - 13,7 )$ 
E)  $\sqrt { 31 } ( 11 , - 13,7 )$ 
A)  $\frac { \sqrt { 31 } } { \sqrt { 339 } } ( 11 , - 13,7 )$ 
B)  $\frac { 1 } { \sqrt { 339 } } ( 11 , - 13,7 )$ 
C)  $( 11 , - 13,7 )$ 
D)  $\sqrt { 339 } ( 11 , - 13,7 )$ 
E)  $\sqrt { 31 } ( 11 , - 13,7 )$

Question 4

Find the vector z,given u = $( 9,4 , - 1 )$ and v = $( 7 , - 2 , - 2 )$ ,and w = $( 35 , - 45 , - 23 )$ . -2u + 5v + 3z = w

Accepted Answer

A)  $( - 9 , - 5,6 )$ 
B)  $( 6,6,6 )$ 
C)  $( 6,9 , - 5 )$ 
D)  $( - 6 , - 9 , - 5 )$ 
E)  $( 6 , - 9 , - 5 )$ 
A)  $( - 9 , - 5,6 )$ 
B)  $( 6,6,6 )$ 
C)  $( 6,9 , - 5 )$ 
D)  $( - 6 , - 9 , - 5 )$ 
E)  $( 6 , - 9 , - 5 )$

Question 5

Find a unit vector in the direction of the vector described below. Initial point: (-9,3,-2)
Terminal point: (4,-7,3)

Accepted Answer

A)  $\sqrt { 294 } ( 13 , - 10,5 )$ 
B)  $\frac { 1 } { \sqrt { 28 } } ( 13 , - 10,5 )$ 
C)  $\sqrt { 28 } ( 13 , - 10,5 )$ 
D)  $( 13 , - 10,5 )$ 
E)  $\frac { 1 } { \sqrt { 294 } } ( 13 , - 10,5 )$ 
A)  $\sqrt { 294 } ( 13 , - 10,5 )$ 
B)  $\frac { 1 } { \sqrt { 28 } } ( 13 , - 10,5 )$ 
C)  $\sqrt { 28 } ( 13 , - 10,5 )$ 
D)  $( 13 , - 10,5 )$ 
E)  $\frac { 1 } { \sqrt { 294 } } ( 13 , - 10,5 )$

Question 6

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector.[Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.] (-4,5,6),n = 2i- 4j+ 4k

Accepted Answer

A) x - 2y + 2 z + 2 = 0 
B) 4x - 5y - 6z + 4 = 0 
C) x - 2y + 2 z - 2 = 0 
D) x - 2y + 2 z = 0 
E) 4x - 5y - 6z - 4 = 0 
A) x - 2y + 2 z + 2 = 0 
B) 4x - 5y - 6z + 4 = 0 
C) x - 2y + 2 z - 2 = 0 
D) x - 2y + 2 z = 0 
E) 4x - 5y - 6z - 4 = 0

Question 7

Find the vector z,given $\mathbf { u } = ( - 1,3,3 ) , \mathbf { v } = ( 1 , - 3 , - 3 )$ .  $\mathbf { z } = \mathbf { u } - 3 \mathbf { v }$

Accepted Answer

A)  $\mathbf { z } = ( - 4,12,12 )$ 
B)  $\mathbf { z } = ( 12 , - 4,12 )$ 
C)  $\mathbf { z } = ( - 4,12 , - 4 )$ 
D)  $\mathbf { z } = ( 12,12,12 )$ 
E)  $\mathbf { z } = ( 12,12 , - 4 )$ 
A)  $\mathbf { z } = ( - 4,12,12 )$ 
B)  $\mathbf { z } = ( 12 , - 4,12 )$ 
C)  $\mathbf { z } = ( - 4,12 , - 4 )$ 
D)  $\mathbf { z } = ( 12,12,12 )$ 
E)  $\mathbf { z } = ( 12,12 , - 4 )$

Question 8

Determine whether the planes are parallel,orthogonal,or neither.  $\begin{array} { l } 
x - 9 y - z = 5 \
4 x - 36 y - 4 z = - 2
\end{array}$

Accepted Answer

A) Neither 
B) Orthogonal 
C) Parallel 
A) Neither 
B) Orthogonal 
C) Parallel

Question 9

Find the general form of the equation of the plane with the given characteristics. The plane passes through the point (-2,-3,-5)and is parallel to the yz-plane.

Accepted Answer

A) x + y + z = -10 
B) y = -3 
C) z = -5 
D) y + z = -8 
E) x = -2 
A) x + y + z = -10 
B) y = -3 
C) z = -5 
D) y + z = -8 
E) x = -2

Question 10

Find the midpoint of the line segment joining the points. (-1,0,-7), (-8,-5,-6)

Accepted Answer

A)  $( - 9,5 , - 13 )$ 
B)  $( - 9 , - 9 , - 9 )$ 
C)  $( - 9 , - 5 , - 13 )$ 
D)  $\left( \frac { 7 } { 4 } , \frac { - 5 } { 4 } , \frac { - 13 } { 4 } \right)$ 
E)  $\left( \frac { - 9 } { 2 } , \frac { - 5 } { 2 } , \frac { - 13 } { 2 } \right)$ 
A)  $( - 9,5 , - 13 )$ 
B)  $( - 9 , - 9 , - 9 )$ 
C)  $( - 9 , - 5 , - 13 )$ 
D)  $\left( \frac { 7 } { 4 } , \frac { - 5 } { 4 } , \frac { - 13 } { 4 } \right)$ 
E)  $\left( \frac { - 9 } { 2 } , \frac { - 5 } { 2 } , \frac { - 13 } { 2 } \right)$

Question 11

Find the area of the parallelogram that has the vectors as adjacent sides.  $\mathbf { u } = ( 3 ) \mathbf { i } + ( 3 ) \mathbf { j } + ( - 4 ) \mathbf { k } , \quad \mathbf { v } = ( - 4 ) \mathbf { i } + ( 5 ) \mathbf { j } + ( 1 ) \mathbf { k }$

Accepted Answer

A) 3 square units 
B)  $0$ square units 
C)  $1$ square units 
D) 37 square units 
E)  $\sqrt { 1427 }$ square units 
A) 3 square units 
B)  $0$ square units 
C)  $1$ square units 
D) 37 square units 
E)  $\sqrt { 1427 }$ square units

Question 12

The weight of a crate is 300 newtons.Find the tension in each of the supporting cables shown in the figure.The coordinates of the points A,B,C,and D are given below the figure.Round to the nearest newton.   [Figure not necessarily to scale.] point A = (0,0,-80),point B = (140,0,0),point C = (-50,50,0),point D = (0,-180,0) $\vec { T } + m \vec { g } = 0$

Accepted Answer

A) cable AB = 245;cable AC = 132;cable AD = 126 
B) cable AB = 132;cable AC = 126;cable AD = 245 
C) cable AB = 132;cable AC = 245;cable AD = 126 
D) cable AB = 126;cable AC = 245;cable AD = 132 
E) cable AB = 245;cable AC = 126;cable AD = 132 
A) cable AB = 245;cable AC = 132;cable AD = 126 
B) cable AB = 132;cable AC = 126;cable AD = 245 
C) cable AB = 132;cable AC = 245;cable AD = 126 
D) cable AB = 126;cable AC = 245;cable AD = 132 
E) cable AB = 245;cable AC = 126;cable AD = 132

Question 13

Find a unit vector in the direction of u.  $\mathbf { u } = - 5 \mathbf { i } + 7 \mathbf { j } + 14 \mathbf { k }$

Accepted Answer

A)  $\frac { u } { \| \mathbf { u } \| } =$ $\frac { \sqrt { 270 } } { 270 } ( - 5 \mathbf { i } + 7 \mathbf { j } + 14 \mathbf { k } )$ 
B)  $\frac { u } { \| \mathbf { u } \| } =$ $- \frac { \sqrt { 270 } } { 270 } ( - 5 \mathbf { i } + 7 \mathbf { j } + 14 \mathbf { k } )$ 
C)  $\frac { u } { \| \mathbf { u } \| } =$ $\frac { 270 } { \sqrt { 270 } } ( - 5 \mathbf { i } + 7 \mathbf { j } + 14 \mathbf { k } )$ 
D)  $\frac { u } { \| \mathbf { u } \| } =$ $\frac { 270 } { 270 } ( - 5 \mathbf { i } + 7 \mathbf { j } + 14 \mathbf { k } )$ 
E)  $\frac { u } { \| \mathbf { u } \| } =$ $- \frac { 270 } { 270 } ( - 5 \mathbf { i } + 7 \mathbf { j } + 14 \mathbf { k } )$ 
A)  $\frac { u } { \| \mathbf { u } \| } =$ $\frac { \sqrt { 270 } } { 270 } ( - 5 \mathbf { i } + 7 \mathbf { j } + 14 \mathbf { k } )$ 
B)  $\frac { u } { \| \mathbf { u } \| } =$ $- \frac { \sqrt { 270 } } { 270 } ( - 5 \mathbf { i } + 7 \mathbf { j } + 14 \mathbf { k } )$ 
C)  $\frac { u } { \| \mathbf { u } \| } =$ $\frac { 270 } { \sqrt { 270 } } ( - 5 \mathbf { i } + 7 \mathbf { j } + 14 \mathbf { k } )$ 
D)  $\frac { u } { \| \mathbf { u } \| } =$ $\frac { 270 } { 270 } ( - 5 \mathbf { i } + 7 \mathbf { j } + 14 \mathbf { k } )$ 
E)  $\frac { u } { \| \mathbf { u } \| } =$ $- \frac { 270 } { 270 } ( - 5 \mathbf { i } + 7 \mathbf { j } + 14 \mathbf { k } )$

Question 14

Determine the octant(s)in which (x,y,z)is located so that the conditions are satisfied. x > 0,y  0

Accepted Answer

A) octant IV 
B) octant III 
C) octant VI 
D) octant V 
E) octant I or octant II 
A) octant IV 
B) octant III 
C) octant VI 
D) octant V 
E) octant I or octant II

Question 15

Find the angle $\theta$ between the vectors.Round your answer to two decimal places.  $\begin{array} { l } 
\mathbf { u } = 14 \mathbf { i } + 56 \mathbf { j } \
\mathbf { v } = - 7 \mathbf { j } + 12 \mathbf { k }
\end{array}$

Accepted Answer

A)  $\Theta \approx 120.26 ^ { \circ }$ 
B)  $\Theta \approx 118.26 ^ { \circ }$ 
C)  $\Theta \approx 121.26 ^ { \circ }$ 
D)  $\Theta \approx 117.26 ^ { \circ }$ 
E)  $\Theta \approx 119.26 ^ { \circ }$ 
A)  $\Theta \approx 120.26 ^ { \circ }$ 
B)  $\Theta \approx 118.26 ^ { \circ }$ 
C)  $\Theta \approx 121.26 ^ { \circ }$ 
D)  $\Theta \approx 117.26 ^ { \circ }$ 
E)  $\Theta \approx 119.26 ^ { \circ }$

Question 16

Find the area.  $A ( 5,5,5 ) , B ( 6,7,8 ) , C ( 10,9,6 ) , D ( 11,11,9 )$

Accepted Answer

A)  $\text { Area } = \sqrt { 242.00 } \text { square units }$ 
B)  $\text { Area } = \sqrt { 332.00 } \text { square units }$ 
C)  $\text { Area } = \sqrt { 122.00 } \text { square units }$ 
D)  $\text { Area } = \sqrt { 302.00 } \text { square units }$ 
E)  $\text { Area } = \sqrt { 260.00 } \text { square units }$ 
A)  $\text { Area } = \sqrt { 242.00 } \text { square units }$ 
B)  $\text { Area } = \sqrt { 332.00 } \text { square units }$ 
C)  $\text { Area } = \sqrt { 122.00 } \text { square units }$ 
D)  $\text { Area } = \sqrt { 302.00 } \text { square units }$ 
E)  $\text { Area } = \sqrt { 260.00 } \text { square units }$

Find the area of the parallelogram that has the vectors as adjacent sides. =+3+3 =+

Find u × v and show that it is orthogonal to both u and v. $\mathbf { u } = ( 10,0,3 ) \quad \mathbf { v } = ( 10,0,0 )$

Find a unit vector in the direction of u. u = $( 11 , - 13,7 )$

Find the vector z,given u = $( 9,4 , - 1 )$ and v = $( 7 , - 2 , - 2 )$ ,and w = $( 35 , - 45 , - 23 )$ . -2u + 5v + 3z = w

Find a unit vector in the direction of the vector described below. Initial point: (-9,3,-2) Terminal point: (4,-7,3)

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector.[Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.] (-4,5,6),n = 2i- 4j+ 4k

Find the vector z,given $\mathbf { u } = ( - 1,3,3 ) , \mathbf { v } = ( 1 , - 3 , - 3 )$ . $\mathbf { z } = \mathbf { u } - 3 \mathbf { v }$

Determine whether the planes are parallel,orthogonal,or neither. x-9y-z=5 4x-36y-4z=-2

Find the general form of the equation of the plane with the given characteristics. The plane passes through the point (-2,-3,-5)and is parallel to the yz-plane.

Find the midpoint of the line segment joining the points. (-1,0,-7), (-8,-5,-6)

Find the area of the parallelogram that has the vectors as adjacent sides. $\mathbf { u } = ( 3 ) \mathbf { i } + ( 3 ) \mathbf { j } + ( - 4 ) \mathbf { k } , \quad \mathbf { v } = ( - 4 ) \mathbf { i } + ( 5 ) \mathbf { j } + ( 1 ) \mathbf { k }$

Find a unit vector in the direction of u. $\mathbf { u } = - 5 \mathbf { i } + 7 \mathbf { j } + 14 \mathbf { k }$

Determine the octant(s)in which (x,y,z)is located so that the conditions are satisfied. x > 0,y < 0

Find the angle $\theta$ between the vectors.Round your answer to two decimal places. =14+56 =-7+12

Find the area. $A ( 5,5,5 ) , B ( 6,7,8 ) , C ( 10,9,6 ) , D ( 11,11,9 )$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Limits and An Introduction To Calculus

Filters

Exam 11: Analytic Geometry In Three Dimensions

Find the area of the parallelogram that has the vectors as adjacent sides. =+3+3 =+

Find u × v and show that it is orthogonal to both u and v. u=(10,0,3)v=(10,0,0)\mathbf { u } = ( 10,0,3 ) \quad \mathbf { v } = ( 10,0,0 )u=(10,0,3)v=(10,0,0)

Find a unit vector in the direction of u. u = (11,−13,7)( 11 , - 13,7 )(11,−13,7)

Find the vector z,given u = (9,4,−1)( 9,4 , - 1 )(9,4,−1) and v = (7,−2,−2)( 7 , - 2 , - 2 )(7,−2,−2) ,and w = (35,−45,−23)( 35 , - 45 , - 23 )(35,−45,−23) . -2u + 5v + 3z = w

Find a unit vector in the direction of the vector described below. Initial point: (-9,3,-2) Terminal point: (4,-7,3)

Find the general form of the equation of the plane passing through the point and perpendicular to the specified vector.[Be sure to reduce the coefficients in your answer to lowest terms by dividing out any common factor.] (-4,5,6),n = 2i- 4j+ 4k

Find the vector z,given u=(−1,3,3),v=(1,−3,−3)\mathbf { u } = ( - 1,3,3 ) , \mathbf { v } = ( 1 , - 3 , - 3 )u=(−1,3,3),v=(1,−3,−3) . z=u−3v\mathbf { z } = \mathbf { u } - 3 \mathbf { v }z=u−3v

Determine whether the planes are parallel,orthogonal,or neither. x-9y-z=5 4x-36y-4z=-2

Find the general form of the equation of the plane with the given characteristics. The plane passes through the point (-2,-3,-5)and is parallel to the yz-plane.

Find the midpoint of the line segment joining the points. (-1,0,-7), (-8,-5,-6)

Find a unit vector in the direction of u. u=−5i+7j+14k\mathbf { u } = - 5 \mathbf { i } + 7 \mathbf { j } + 14 \mathbf { k }u=−5i+7j+14k

Determine the octant(s)in which (x,y,z)is located so that the conditions are satisfied. x > 0,y < 0

Find the angle θ\thetaθ between the vectors.Round your answer to two decimal places. =14+56 =-7+12

Find the area. A(5,5,5),B(6,7,8),C(10,9,6),D(11,11,9)A ( 5,5,5 ) , B ( 6,7,8 ) , C ( 10,9,6 ) , D ( 11,11,9 )A(5,5,5),B(6,7,8),C(10,9,6),D(11,11,9)

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Analytic Trigonometry

Additional Topics In Trigonometery

Systems Of Equations and Inequalities

Matrices and Determinants

Sequences Series and Probability

Topics In Analytic Geometry

Limits and An Introduction To Calculus

Filters

Find u × v and show that it is orthogonal to both u and v. $\mathbf { u } = ( 10,0,3 ) \quad \mathbf { v } = ( 10,0,0 )$

Find a unit vector in the direction of u. u = $( 11 , - 13,7 )$

Find the vector z,given u = $( 9,4 , - 1 )$ and v = $( 7 , - 2 , - 2 )$ ,and w = $( 35 , - 45 , - 23 )$ . -2u + 5v + 3z = w

Find the vector z,given $\mathbf { u } = ( - 1,3,3 ) , \mathbf { v } = ( 1 , - 3 , - 3 )$ . $\mathbf { z } = \mathbf { u } - 3 \mathbf { v }$

Find a unit vector in the direction of u. $\mathbf { u } = - 5 \mathbf { i } + 7 \mathbf { j } + 14 \mathbf { k }$

Find the angle $\theta$ between the vectors.Round your answer to two decimal places. =14+56 =-7+12

Find the area. $A ( 5,5,5 ) , B ( 6,7,8 ) , C ( 10,9,6 ) , D ( 11,11,9 )$