Exam 9: Vectors in Two and Three Dimensions

Find the vector with initial point $P ( 4,2 )$ and terminal point $Q ( 3,5 )$ .

Find $\mathbf { u} \cdot \mathbf { v }$ and the angle between $\mathbf { u }$ and $\mathbf { v }$ for $\mathbf { u } = \langle - 5,3 \rangle , \mathbf { v } = \langle \sqrt { 2 } , 7 \rangle$ .

Describe the trace of the sphere $( x + 1 ) ^ { 2 } + ( y - 3 ) ^ { 2 } + ( z + 8 ) ^ { 2 } = 100$ in (a) the yz-plane and (b) the plane $x = 4$ .

A plane has normal vector n and passes through the point P. Find an equation for the plane. $\mathbf { n } = 5 \mathbf { i } - \mathbf { j } + 2 \mathbf { k } , P ( 0,2 , - 3 )$

Three vectors a, b, and c are given. (a) Find their scalar triple product $\mathbf { a } \cdot ( \mathbf { b } \times \mathbf { c } )$ . (b) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine. $\mathbf { a } = \langle 3,0 , - 4 \rangle , \mathbf { b } = \langle 2,2,2 \rangle , \mathbf { c } = \langle 7,4,0 \rangle$

Find the component of $\mathbf { u }$ along $\mathbf { v }$ if $\mathbf { u } = \langle 0,7 \rangle$ and $v = ( 6,8 )$ .

A man pulls a sled horizontally by exerting $32$ lb on the rope that is tied to its front end. If the rope makes an angle of $45 ^ { \circ }$ with horizontal, find the work done in moving the sled $55$ ft.

Find the magnitude and direction (in degrees) of the vector $\mathbf { v } = - 3 \mathbf { i } + 2 \mathbf { j }$ .

Given $\mathbf { u }$ and $\mathbf { v }$ in the figure, sketch $2 \mathbf { u } - \mathbf { v }$ .

For the given vectors a and b, find the cross product $\mathbf { a } \times \mathbf { b }$ . $\mathbf { a } = \langle 0 , - 4,2 \rangle , \mathbf { b } = \langle 1 , - 1 , - 2 \rangle$

Find the work done by the force $\mathbf { F } = 5 \mathbf { i } + 25 \mathbf { j }$ in moving an object from $P ( 0,0 )$ to $Q ( 1,6 )$ .

A jet is flying in a direction N $45 ^ { \circ }$ E with a speed of $430$ mi $/$ h, find the north and east components of the velocity.

Find the vector with initial point $( - 3 , - 9 )$ and terminal point $( - 3,9 )$ .

Find the component of $\mathbf { u }$ along $\mathbf { v}$ if $\mathbf { u } = - \frac { 1 } { 2 } \mathbf { i } + 2 \mathbf { j }$ and $\mathbf { v } = - 2 \mathbf { i } + \frac { 1 } { 2 } \mathbf { j }$ .

Find $| v |$ and $| \mathbf { u } + \mathbf { v } |$ , given that $\mathbf { u } = \langle 2,3 \rangle$ and $\mathbf { v } = \langle - 1,2 \rangle$ .

Find the magnitude and direction (in degrees) of the vector $\mathbf { v } = - \mathbf { i } - \sqrt { 5 } \mathbf { j }$ .

Find $\mathbf { u} \cdot \mathbf { v }$ and the angle between $\mathbf { u }$ and $\mathbf { v }$ for $\mathbf { u } = \sqrt { 2 } \mathbf { i } + 2 \mathbf { j }$ , $\mathbf { v } = - 4 \mathbf { i } - 7 \mathbf { j }$ .

Find the magnitude of the given vector. $\langle - 1,6 , \sqrt { 2 } \rangle$

Find parametric equations for the line that passes through the point P and is parallel to the vector v. $P ( 0,0,0 ) , \quad \mathbf { v } = \langle - 5,2,9 \rangle$

Find an equation of a sphere with the given radius r and center C. $r = \sqrt { 10 }$ ; $C ( - 10,0,1 )$