Exam 9: Vectors in Two and Three Dimensions

Express the given vector in terms of the unit vectors i, j, and k. $\langle 0,3 , - 5 \rangle$

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

Find an equation of a sphere with the given radius r and center C. $r = 4$ ; $C ( 2 , - 5,3 )$

Find the horizontal and vertical components of the vector with length $| \mathbf { v } | = 240$ , and direction $\theta = 300 ^ { \circ }$ .

A car is on a ramp that is inclined $14 ^ { \circ }$ to the horizontal. If the car weighs $2235$ lb, find the force required to keep the car from rolling down the ramp.

Find the vector with initial point $P ( 5,1 )$ and terminal point $Q ( 2,2 )$ .

The lengths of two vectors a and b, and the angle $\theta$ between them, are given. Find the length of their cross product, $| \mathbf { a } \times \mathbf { b } |$ . $| \mathbf { a } | = 4 , |\mathbf { b }| = 5 , \theta = 60 ^ { \circ }$

Find the vector with initial point $P ( 7,3 )$ and terminal point $Q ( 1,4 )$ .

Find an equation of a sphere with the given radius r and center C. $r = 5$ ; $C ( 0,5 , - 3 )$

Describe the surface represented by the given equation. $z = 7$

If the vector v has initial point P, what is its terminal point? $\mathbf { v } = \langle 0,0,1 \rangle , P ( 1,1 , - 1 )$

Find the area of the triangle V $P Q R$ . $P ( 3 , - 2,6 ) , Q ( - 1 , - 4 , - 6 ) , R ( 3,3,6 )$

Find the work done by the force $F$ $= 100 \mathbf { i } - 25 \mathbf { j }$ in moving an object from $P ( - 2,4 )$ to $Q ( 150,5 )$ .

Describe the surface represented by the given equation. $y = 1$

Determine whether $\mathbf { u } = 2 \sqrt { 3 } \mathbf { i } + 2 \mathbf { j }$ is orthogonal to $\mathbf { v } = \mathbf { i } - \sqrt { 3 } \mathbf { j }$ .

A water tank is in the shape of a sphere of radius 10 feet. The tank is supported on a metal circle 8 feet below the center of the sphere, as shown in the figure. Find the radius of the metal circle.

Find $\mathbf {u } \cdot \mathbf { v }$ and the angle between $\mathbf { u }$ and $\mathbf { v }$ for $\mathbf { u } = - 6 \mathbf { j }$ , $\mathbf { v } = - 2 \mathbf { i } + 2 \sqrt { 3 } \mathbf { j }$ .

Find parametric equations for the line that passes through the points P and Q. $P ( 10,16,18 ) , \quad Q ( 10 , - 6,0 )$

Find parametric equations for the line that passes through the point P and is parallel to the vector v. $P ( 2,2,2 ) , \quad \mathbf { v } = \mathbf { i } - \mathbf { j } + \mathbf { k }$

Find the magnitude and direction (in degrees) of the vector $\mathbf { v } = \langle 5,12 \rangle$ .