91Ó°ÊÓ

When the direction numbers \(a , b ,\) and \(c\) of the vector \(\mathbf { v } = \langle a , b , c )\) are all nonzero, you can elimininate the parameter to obtain the ____ ____ of a line.

If the dot product of two nonzero vectors is zero, then the angle between the vectors is \(90^{\circ}\) and the vectors are called _______.

The dot product of \(\mathbf{u}\) and \(\mathbf{v} \times \mathbf{w}\) is called \(t\)______ _______ ______of \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\)

The distance between the points \(\left(x_{1}, y_{1}, z_{1}\right)\) and \(\left(x_{2}, y_{2}, z_{2}\right)\) can be found using the_____________ _____________ in Space.

A vector that is perpendicular to a plane is called ______.

Two nonzero vectors \(\mathbf{u}\) and \(\mathbf{v}\) are ________ when there is some scalar \(c\) such that \(\mathbf{u}=c \mathbf{v}\).

Finding Equations In Exercises \(5 - 10\) , find (a) a set of parametric equations and (b) if possible, a set of symmetric equations of the line passing through the point and parallel to the specified vector or line. (Write the direction numbers as integers.) $$ \begin{array} { l l } { \text { Point } } & { \text { Parallel to } } \\ { ( 0,0,0 ) } & { \mathbf { v } = \langle 1,2,3 \rangle } \end{array} $$

The midpoint of the line segment joining the points \(\left(x_{1}, y_{1}, z_{1}\right)\) and \(\left(x_{2}, y_{2}, z_{2}\right)\) given by the Midpoint Formula in Space is________________.

Performing Vector Operations In Exercises \(5-14\) use the vectors \(u=3 i-j+4 k\) and \(v=2 i+2 j-k\) to find the expression. $$\mathbf{u} \times \mathbf{v}$$

A_____________is the set of all points \((x, y, z)\) such that the distance between \((x, y, z)\) and a fixed point \((h, k, j)\) is \(r\).