Exam 9: Vectors in Two and Three Dimensions

Find $\mathbf { u } \cdot \mathbf { v }$ and the angle between $\mathbf { u }$ and $\mathbf { v }$ for $\mathbf { u } = 2 \mathbf { i } + 2 \mathbf { j } , \mathbf { v } = - 4 \mathbf { i } - 4 \mathbf { j }$ .

Two vectors u and v are given. Find the angle (expressed in degrees) between u and v. $\mathbf { u } = \langle 4,0 , - 2 \rangle , \mathbf { v } = \langle 2,1,0 \rangle$

A plane has normal vector n and passes through the point P. Find an equation for the plane. $\mathbf { n } = \langle3,0 , - \frac { 1 } { 3 } \rangle , P ( 2,4,9 )$

Find the vector v with initial point P and terminal point Q. $P ( 1 , - 1 , - 1 )$ , $Q ( - 1,0 , - 1 )$

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 19 = 12 x - 16 z$

Find a vector that is perpendicular to the plane passing through the three given points. $P ( 3,4,5 ) , Q ( 1,0,3 ) , R ( 4,7,6 )$

Find the magnitude and direction (in degrees) of the vector $\mathbf { v } = \langle \sqrt { 3 } , 1 \rangle$ .

Determine whether $\mathbf { u } = \langle - 3,0 \rangle$ is orthogonal to $\mathbf { v } = \langle 0 , - 5 \rangle$ .

Given $\mathbf { u } = \langle 10,5 \rangle$ and $\mathbf { v } = \langle 3,4 \rangle$ , calculate $\operatorname { proj } _ { \mathbf { v } } \mathbf { u }$ and then resolve $\mathbf { u }$ into $\mathbf { u } _ { 1 }$ and $\mathbf { u } _ { 2 }$ .

Find the vector with initial point $P ( 2,4 )$ and terminal point $Q ( - 3,1 )$ .

Describe the surface represented by the given equation. $y = - 3$

Find the magnitude and direction (in degrees) of the vector $\mathbf { v } = \mathbf { i } - \mathbf { j }$ .

Express the given vector in terms of the unit vectors i, j, and k. $\langle - a , - \frac { 1 } { 2 } a , 4 \rangle$

Find $\mathbf {u} \cdot \mathbf { v }$ and the angle between $\mathbf { u }$ and $\mathbf { v }$ for $\mathbf { u } = \langle 5,0 \rangle , \mathbf { v } = \langle \sqrt { 3 } , 1 \rangle$ .

Given $\mathbf { u }$ and $\mathbf { v}$ in the figure, sketch $\mathbf { u } + \mathbf { v }$ .

Two vectors u and v are given. Find their dot product $\mathbf { u} \cdot \mathbf { v }$ . $\mathbf { u } = 4 \mathbf { j } - 2 \mathbf { k } , \mathbf { v } = \frac { 5 } { 6 } \mathbf { i } - \frac { 5 } { 4 } \mathbf { j }$