91Ó°ÊÓ

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

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.) $$ ( - 2,0,3 ) \quad \mathbf { v } = 2 \mathbf { i } + 4 \mathbf { j } - 2 \mathbf { k } $$

Determining Octants In Exercises \(21-26,\) determine the octant(s) In which \((x, y, z)\) is located so that the condition(s) is (are) satisfied. $$z>0$$

Determining Octants In Exercises \(21-26,\) determine the octant(s) In which \((x, y, z)\) is located so that the condition(s) is (are) satisfied. $$y z>0$$

Finding an Equation of a Plane in Three-Space In Exercises \(31 - 36\) , find the general form of the equation of the plane with the given characteristics. $$ \begin{array} { l } { \text { Passes through } ( 1 , - 2,4 ) \text { and } ( 4,0 , - 1 ) \text { and is } } \\ { \text { perpendicular to the } x z \text { -plane } } \end{array} $$

Finding the Area of a Triangle In Exercises \(49-52\) , find the area of the triangle with the given vertices. $$(2,4,0),(-2,-4,0),(0,0,4)$$

Finding the Terminal Point of a Vector In, the vector \(v\) and its initial point are given. Find the terminal point. $$\begin{array}{l}{\mathbf{v}=\langle 4,-1,-1\rangle} \\ {\text { Initial point: }(6,-4,3)}\end{array}$$

Finding a Vector, find the component form of v. $$\begin{array}{l}{\text { Vector } \mathbf{v} \text { lies in the } y z-\text { plane, has magnitude } 4, \text { and }} \\ {\text { makes an angle of } 45^{\circ} \text { with the positive } y \text { -axis. }}\end{array}$$

Finding a Vector, find the component form of v. $$\begin{array}{l}{\text { Vector } \mathbf{v} \text { lies in the } x z \text { -plane, has magnitude } 10, \text { and }} \\ {\text { makes an angle of } 60^{\circ} \text { with the positive } z \text { -axis. }}\end{array}$$

Finding the Distance Between a Point and a Plane In Exercises \(59 - 66 ,\) find the distance between the point and the plane. $$ \begin{array} { l } { ( 4 , - 2 , - 2 ) } \\ { 2 x - y + z = 4 } \end{array} $$