Exam 9: Vectors and the Geometry of Space

Given points A = $\left( 2 , \frac { 5 \pi } { 6 } , 1 \right)$ and B = $\left( 3 , \frac { \pi } { 2 } , - 2 \right)$ in cylindrical coordinates, find the distance between the two points.

Find the distance from the point (3, 3, 5) to the point (0, -1, -1).

Find an equation of the plane whose graph is given below.

Find the length of the vector $\mathbf { a } - \mathbf { b }$ , where $\mathbf { a } = \langle 1,3 \rangle$ and $\mathbf { b } = \langle 5,2 \rangle$ .

Find an equation of the plane containing the lines $\frac { x } { 2 } = \frac { y - 2 } { 3 } = \frac { z + 1 } { 1 }$ and $\frac { x + 1 } { 4 } = \frac { y } { 6 } = \frac { z - 3 } { 2 }$

The center of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 4 x - 2 y - 6 z = 0$ is

Let a and b be unit vectors and c = a + b. Use the dot product to prove that the angle between a and c is equal the angle between b and c.

Which two of the following four vectors are parallel? (a) $( 0,8,2 )$ (b) $( - 1,4,1 )$ (c) $\left( - \frac { 1 } { 2 } , 2 , \frac { 1 } { 2 } \right)$ (d) $\left( \frac { 7 } { 2 } , 5 , - \frac { 7 } { 2 } \right)$

Find the coordinate of the initial point of a vector $\mathbf { a } = \langle 3 , - 2,2 \}$ whose terminal point is $( 4 , - 3,5 )$ .

The lines $l _ { 1 } :$ $\mathbf { r } _ { 1 } = \langle 2,3,1 \rangle+ t \langle - 1,1,1 \rangle$ and l₂: $\mathbf { r } _ { 2 } = \langle 3,1,2 \rangle + s \langle - 1,2 , - 1\rangle$ both contain the point P (2, 3, 1). Find the value of s which gives the point of intersection P, and then compute the angle $\theta$ between the two lines.

Let P = (1, 3, 2) and let L be the line with parametric equations x = 2 - t, y = - 1 + 2t, z = 3 + t. Use the vector cross product to find the distance from P to L.

Use the property of the cross product that $| \mathbf { u } \times \mathbf { v } | = | \mathbf { u } | | \mathbf { v } | \sin \theta$ to derive a formula for the distance d from a point P to a line l. Use this formula to find the distance from the origin to the line through (2, 1, -4) and (3, 3, -2).

Identify the surface $- x ^ { 2 } + y ^ { 2 } - z ^ { 2 } = 10$ .

Find two unit vectors that are orthogonal to both $\langle 1 , - 1,1 \rangle$ and $\langle0,2,2 \rangle$ .

Find an equation of a plane containing the points P(2, -1, 1), Q(5, 0, -1) and R(-2, 3, 3).

Find a unit vector orthogonal to both of the vectors $\{ 1 , - 1,0 \}$ and $\{ 1,2,3 \}$ .

Find $| \mathbf { a } |$ , a + b, $\mathbf { a } - \mathbf { b }$ , 2a, and 3a + 4b given $\mathbf { a } = ( 3,2 , - 1 )$ and $\mathrm { b } = \langle 0,6,7 \rangle$ .

Find a unit vector perpendicular to the plane x - 2y - 2z = 10.

Find the length of the vector from the point P(1, 2) to the point Q(3, 4).

Let $\mathbf { u } \times \mathbf { v } = \mathbf { i } - \mathbf { j } + 2 \mathbf { k }$ . Find $\mathbf { u } \cdot ( \mathbf { u } \times \mathbf { v } )$ ), if it exists.