Exam 10: Vectors

What is the distance between the given pair of points: $( 1,2 , - 1 ) \text { and } ( 4,2,2 )$ ?

Find the dot product of $\vec { u } = \langle 4,7 \rangle$ and $\vec { v } = \langle 1,3 \rangle$ and the cosine of the angle between them.

Give the symmetric equations of the line containing the points $( 1,3,2 ) \text { and } ( 2,1,4 )$

Give the symmetric equations of the line containing the points $( 2,2 - 3 ) \text { and } ( 6,1,2 )$

Give an equation of the line containing the points $( 2,1,1 ) \text { and } ( 4,3,2 )$ as vector parameterization. A) $\vec { r } ( t ) = \langle 1 , - 2,2 \rangle + t \langle 1,3,2 \rangle$ B) $\vec { r } ( t ) = \langle 2,1,1 \rangle + t \langle 2,2,1 \rangle$ C) $\vec { r } ( t ) = \langle 1,3,2 \rangle + t \langle 1,2,2 \rangle$ D) $\vec { r } ( t ) = \langle 1,2,2 \rangle + t \langle 1,3,2 \rangle$

Assume the segment with the endpoints $( 1,2,3 )$ and $( 5 , - 4,5 )$ is a diameter of a sphere. Give an equation of the sphere.

Are the given lines parallel, intersecting, identical or skew? $\vec { r } _ { 1 } ( t ) = \langle 2,3,6 \rangle + t \langle - 1,2,4 \rangle$ $\vec { r } _ { 2 } ( t ) = \langle 9 , - 1 , - 2 \rangle + t \langle 2 , - 4 , - 8 \rangle$

If $( 3,3 , - 3 )$ is the midpoint of the segment with one endpoint $( 1,2 , - 2 )$ , find the second endpoint.

Find the area of the triangle in the plane with vertices $P ( 1,2 ) , Q ( 4,6 ) , \text { and } R ( - 1 , - 1 )$

Find the dot product of $\vec { u } = \langle 2,3 \rangle$ and $\vec { v } = \langle - 3,2 \rangle$ and the cosine of the angle between them.

For $\vec { u } = \langle 1,2,3 \rangle$ and $\vec { v } = \langle - 1,1 , - 1 \rangle$ $\vec { w } = \langle 1,2,1 \rangle$ , find $\vec { u } \times \vec { v }$

Find the area of the parallelogram determined by $\vec { v } = \langle - 1,1 , - 1 \rangle$ and $\vec { w } = \langle 1,2,1 \rangle$

Find the center and radius of the sphere with the equation $x ^ { 2 } - 2 x + y ^ { 2 } - 2 y + z ^ { 2 } + 2 z - 6 = 0$

Find $\operatorname { com } p _ { \vec { u } } \vec { v }$ , $\operatorname { proj } _ { \vec { u } } \vec { v }$ , and the component of $\vec { v }$ orthogonal to $\vec { u }$ , where $\vec { u } = \langle 1,1,1 \rangle \text { and } \vec { v } = \langle 3,2,1 \rangle$

Give a set of symmetric equations for the line containing the points $( 2,1,1 ) \text { and } ( 4,3,2 )$ A) $\frac { x - 2 } { 2 } = \frac { y - 1 } { 2 } = z - 1$ B) $x - 1 = \frac { 3 - y } { 2 } = \frac { z - 2 } { 2 }$ C) $x - 1 = \frac { y - 3 } { 2 } = \frac { z - 2 } { 2 }$ D) $x - 1 = \frac { y - 2 } { 3 } = \frac { z - 2 } { 2 }$

Find an equation for the plane containing the point $( - 1,1,3 )$ and normal to the vector $\langle 1,2,4 \rangle$

For $\vec { u } = \langle - 1 , - 1,4 \rangle \text { and } \vec { v } = \langle 1 , - 1,4 \rangle$ find $\vec { u } + \vec { v }$ and $\vec { u } - \vec { v }$

For $\vec { u } = \langle 1,2,3 \rangle$ , $\vec { v } = \langle - 1,1 , - 1 \rangle$ , and $\vec { w } = \langle 1,2,1 \rangle$ , find $\vec { v } \cdot ( \vec { u } \times \vec { w } )$

Find a vector of length 10 parallel to $\langle 3,4 \rangle$

Give the parametric equations of the line containing the points $( 2,2 - 3 ) \text { and } ( 6,1,2 )$