Exam 10: Vectors

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

Find the line of intersection of the two planes given by $x + y + z = 4$ and $2 x - y + z = 3$

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 $\vec { u } = \langle 1,2,1 \rangle \text { and } \vec { v } = \langle - 1,1 , - 1 \rangle$

Find the norm of $\vec { u } = \langle 1,2 \rangle$

For $P = ( - 1,3,9 ) \text { and } Q = ( 7,6,2 )$ , find $\overrightarrow { P Q }$

Find a vector perpendicular to the plane determined by the points $P ( 1,2,3 ) , Q ( 4,5,6 ) \text {, and } R ( 1,1 , - 1 )$ , whose third coordinate is 1.

Find the volume of the parallelepiped determined by $\vec { u } = \langle 1,2,3 \rangle$ , $\vec { v } = \langle - 1,1 , - 1 \rangle$ , and $\vec { w } = \langle 1,2,1 \rangle$

Find the dot product of $\vec { u } = \langle 1,2,1 \rangle$ and $\vec { v } = \langle 2,2,1 \rangle$ and the cosine of the angle between them.

Find an equation for the plane containing the points $( 1,2,3 ) , ( 3,1,1 ) \text {, and } ( - 1,3,2 )$

Find the equation of the sphere with the center $( 1,2,3 )$ containing the origin.

Find the center and radius of the sphere with the equation $x ^ { 2 } - 2 x + y ^ { 2 } - 4 y + z ^ { 2 } + 1 = 0$

Find an equation for the plane containing the lines $\vec { r } _ { 1 } ( t ) = \langle 1,2,3 \rangle + t \langle 1 , - 1,2 \rangle$ $\vec { r } _ { 2 } ( t ) = \langle 0,1,2 \rangle + t \langle - 2,2 , - 4 \rangle$

Give an equation of the line containing the points $( 1,3,2 ) \text { and } ( 2,1,4 )$ as vector parameterization.

Find the area of the triangle with vertices $P ( 0,0,0 ) , Q ( 1,2,3 ) , \text { and } R ( 1,2,1 )$

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

Find an equation for the plane that contains the point $( 1,3,7 )$ and the line $\vec { r } ( t ) = \langle 2,1,0 \rangle + t \langle 1,1,2 \rangle$

Find the center and radius of the sphere with the equation $x ^ { 2 } + 2 x + y ^ { 2 } - 2 y + z ^ { 2 } - 4 z + 5 = 0$

Find an equation for the plane containing the points $( 2,1 , - 1 ) , ( 3,2,7 ) \text {, and } ( 4,3,1 )$

Find the distance from the point $( 1,1,2 )$ to the line $\vec { r } ( t ) = \langle 0 , - 1,3 \rangle + t \langle 1,2,1 \rangle$

For $P = ( 1,2,1 ) \text { and } Q = ( - 4,2,2 )$ , find $\overrightarrow { P Q }$