91Ó°ÊÓ

Finding the Coordinates of a Point In Exercises \(15-20\) , find the coordinates of the point. The point is located three units behind the \(y z\) -plane, four units to the right of the \(x z\) -plane, and five units above the \(x y\) -plane.

Finding the Cross Product In Exercises \(15-28\) , find \(\mathbf{u} \times \mathbf{v}\) and show that it is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v} Finding the Cross Product In Exercises \)15-28\( , find \)\mathbf{u} \times \mathbf{v}\( and show that it is orthogonal to both \)\mathbf{u}\( and \)\mathbf{v} .$ $$\mathbf{u}=\langle 1,-1,0\rangle$$ $$\mathbf{v}=\langle 0,1,-1\rangle$$

Finding Equations In Exercises \(11 - 18 ,\) find (a) a set of parametric equations and (b) if possible, a set of symmetric equations of the line that passes through the points. (Write the direction numbers as integers.) $$ ( 3,1,2 ) , ( - 1,1,5 ) $$

Sketching Vectors, sketch each scalar multiple of v. $$\begin{array}{l}{\mathbf{v}=\langle 1,1,3\rangle} \\ {\text { (a) } 2 \mathbf{v} \quad \text { (b) }-\mathbf{v} \quad \text { (c) } \frac{3}{2} \mathbf{v} \quad \text { (d) } 0 \mathbf{v}}\end{array}$$

Finding the Cross Product In Exercises \(15-28\) , find \(\mathbf{u} \times \mathbf{v}\) and show that it is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v} .\) $$\mathbf{u}=\langle- 1,1,0\rangle$$ $$\mathbf{v}=\langle 1,0,-1\rangle$$

Sketching Vectors, sketch each scalar multiple of v. $$\begin{array}{l}{\mathbf{v}=\langle- 1,2,2\rangle} \\ {\text { (a) }-\mathbf{v} \quad \text { (b) } 2 \mathbf{v} \quad \text { (c) } \frac{1}{2} \mathbf{v} \quad \text { (d) } \frac{5}{2} \mathbf{v}}\end{array}$$

Finding Equations In Exercises \(11 - 18 ,\) find (a) a set of parametric equations and (b) if possible, a set of symmetric equations of the line that passes through the points. (Write the direction numbers as integers.) $$ ( 2 , - 1,5 ) , ( 2,1 , - 3 ) $$

Finding Equations In Exercises \(11 - 18 ,\) find (a) a set of parametric equations and (b) if possible, a set of symmetric equations of the line that passes through the points. (Write the direction numbers as integers.) $$ \left( - \frac { 1 } { 2 } , 2 , \frac { 1 } { 2 } \right) , \left( 1 , - \frac { 1 } { 2 } , 0 \right) $$

Sketching Vectors, sketch each scalar multiple of v. $$\begin{array}{l}{\mathbf{v}=2 \mathbf{i}+2 \mathbf{j}-\mathbf{k}} \\ {\text { (a) } 2 \mathbf{v} \quad \text { (b) }-\mathbf{v} \quad \text { (c) } \frac{5}{2} \mathbf{v} \quad \text { (d) } 0 \mathbf{v}}\end{array} $$

Finding the Cross Product In Exercises \(15-28\) , find \(\mathbf{u} \times \mathbf{v}\) and show that it is orthogonal to both \(\mathbf{u}\) and \(\mathbf{v} .\) $$\mathbf{u}=\langle 2,-3,4\rangle$$ $$\mathbf{v}=\langle 0,-1,1\rangle$$