Exam 11: Vectors; Lines, Planes, and Quadric Surfaces in Space

The value of a that makes $\mathbf { v } = - 2 \mathbf { i } + \mathbf { j } - 4 \mathbf { k }$ and $\mathbf { w } = 3 \mathbf { i } - \mathbf { j } + a \mathbf { k }$ orthogonal is

Which of the following is not true? ​

Let , and $\mathbf { u } = \langle 1,1 , - 6 , \rangle$ Then $\mathbf { v } = \langle 3,0,8 \rangle$ is

The cross product of $\mathbf { v } = 2 \mathbf { i } - 3 \mathbf { j } - 8 \mathbf { k }$ and $\mathbf { w } = \mathbf { i } + 6 \mathbf { k }$ is

In the three-dimensional Euclidean space, the graph of $\frac { x ^ { 2 } } { 9 } - y ^ { 2 } + \frac { z ^ { 2 } } { 16 } = 1$ is

The cross product of $\mathbf { v } = 2 \mathbf { i } + 3 \mathbf { j } + \mathbf { k }$ and $\mathbf { w } = \mathbf { i } - \mathbf { j } - \mathbf { k }$ is

The area of the parallelogram PQRS with vertices $P = ( - 2,1,6 ) , Q = ( 2,1 , - 7 )$ and $R = ( 1,5 , - 4 )$ is

The value of a that makes $\mathbf { v } = 2 \mathbf { i } - \mathbf { j } + 3 \mathbf { k }$ and $\mathbf { w } = 4 \mathbf { i } + a \mathbf { j } + \mathbf { k }$ orthogonal is

The equation of the plane through (4,0,5) parallel to the xz-plane is ​

Let $P _ { 1 } = ( 1,3,5 )$ and $P _ { 2 } = ( - 2,4 , - 6 )$ v is $\overrightarrow { P _ { 1 } P _ { 2 } }$ in standard position, then v is

The unit vector resulted from normalizing the vector $\mathbf { w } = \langle 2 , - 1,4 \rangle$ is

The cross product of $\mathbf { v } = \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k }$ and $4 \mathbf { i } - \mathbf { j } + 3 \mathbf { k }$ is

The vector projection of $\mathbf { v } = - 3 \mathbf { i } + \mathbf { j } - \mathbf { k }$ onto $\mathbf { w } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k }$ is

Which of the following is not true? ​

The distance from the point (2,2,3) to the plane $2 x + y + 2 z = 4$ is

The distance between $P _ { 1 } = ( 3,2,6 )$ and $P _ { 2 } = ( - 1,2,5 )$ is

The area of the parallelogram PQRS with vertices $P = ( - 2,1,6 ) , Q = ( 2,1 , - 7 )$ and $R = ( 4,1,1 )$ is

Which of the following is not a scalar? ​

The center and radius of the sphere $x ^ { 2 } + y ^ { 2 } + z ^ { 2 } + 2 x - 2 y = 2$ are

The center and radius of the sphere $3 x ^ { 2 } + 3 y ^ { 2 } + 3 z ^ { 2 } + 6 x - z - 3 = 0$ are