Exam 9: Vectors in Two and Three Dimensions

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

For the given vectors a and b, find the cross product $\mathbf { a } \times \mathbf { b }$ . $\mathbf { a } = \mathbf { i } + \mathbf { j } + \mathbf { k } , \mathbf { b } = 3 \mathbf { i } - 5 \mathbf { k }$

Find the area of the triangle V $P Q R$ . $P ( 4,1,0 ) , Q ( 0,0 , - 1 ) , R ( - 8,2,0 )$

Find the magnitude of the given vector. $\langle - 5,0,12 \rangle$

Find an equation of the plane that passes through the points P, Q, and R. $P ( 8,1,1 ) , Q ( 5,2,0 ) , R ( 0,0,0 )$

If the vector v has initial point P, what is its terminal point? $\mathbf { v } = \langle 23 , - 5,13 \rangle , P ( - 6,5,2 )$

Determine whether $\mathbf { u } = 8 \mathbf { i } + 4 \mathbf { j }$ is orthogonal to $\mathbf { v } = 6 \mathbf { i } - 12 \mathbf { j }$ .

A jet heads in the direction N $30 ^ { \circ }$ E at a speed of $400$ mi/h. If the jet experiences a $20$ mi/h crosswind flowing due east, find (a) the true velocity $\text { W }$ of the jet. (b) the true speed of the jet. (c) the direction of the jet.

Find the distance between the points $P ( 3,1,0 )$ and $Q ( 3 , - 6,7 )$ .

Find the distance between the points $P ( 5 , - 4 , - 6 )$ and $Q ( - 8 , - 6,5 )$ .

Find the vectors $\mathbf { u } + \mathbf { v }$ , $\mathbf { u } - \mathbf { v }$ , and $3 \mathbf { u } - \frac { 1 } { 2 } \mathbf { v }$ . $\mathbf { u } = 2 \mathbf { i } + \mathbf { j } , \mathbf { v } = - 2 \mathbf { j } - \mathbf { k }$

Find the distance between the points $P ( 5,0,10 )$ and $Q ( 12 , - 3,5 )$ .

Find the center and radius of the sphere. $x ^ { 2 } - 2 x + y ^ { 2 } + 4 y + z ^ { 2 } + 6 z = 211$

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } - 4 x + 4 - 8 y = 0$

Find an equation of the plane that passes through the points P, Q, and R. $P \left( \frac { 3 } { 2 } , 4 , - 3 \right) , Q \left( - \frac { 1 } { 2 } , 2,0 \right) , R \left( - \frac { 1 } { 2 } , 0,3 \right)$

A description of a plane is given. Find an equation for the plane. The plane that is parallel to the plane $x - 2 y + 4 z = 8$ and contains the origin.

A water tank is in the shape of a sphere of radius 15 feet. The tank is supported on a metal circle 12 feet below the center of the sphere, as shown in the figure. Find the radius of the metal circle.

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

Find the center and radius of the sphere. $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 8 y + 2 z = 19$

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