91Ó°ÊÓ

Determining Orthogonal and Parallel Vectors, determine whether \(u\) and \(v\) are orthogonal, parallel, or neither. $$\begin{array}{l}{\mathbf{u}=\langle 0,4,-1\rangle} \\ {\mathbf{v}=\langle 1,0,0\rangle}\end{array}$$

Using the Midpoint Formula in Space In Exercises \(45-52\) , find the midpoint of the line segment joining the points. $$(-1,5,-3),(3,7,-1)$$

Determining Orthogonal and Parallel Vectors, determine whether \(u\) and \(v\) are orthogonal, parallel, or neither. $$\begin{array}{l}{\mathbf{u}=\frac{3}{4} \mathbf{i}-\frac{1}{2} \mathbf{j}+2 \mathbf{k}} \\ {\mathbf{v}=4 \mathbf{i}+10 \mathbf{j}+\mathbf{k}}\end{array}$$

Finding the Area of a Triangle In Exercises \(49-52\) , find the area of the triangle with the given vertices. $$(0,0,0),(1,2,3),(-3,0,0)$$

Finding the Line of Intersection of Two Planes In Exercises \(47 - 52 ,\) (a) find the angle between the two planes and (b) find parametric equations of their line of intersection. $$ \begin{aligned} x - 3 y + z & = - 2 \\ 2 x & \+ 5 z + 3 = & 0 \end{aligned} $$

Finding the Area of a Triangle In Exercises \(49-52\) , find the area of the triangle with the given vertices. $$(1,-4,3),(2,0,2),(-2,2,0)$$

Determining Orthogonal and Parallel Vectors, determine whether \(u\) and \(v\) are orthogonal, parallel, or neither. $$\begin{array}{l}{\mathbf{u}=-\mathbf{i}+\frac{1}{2} \mathbf{j}-\mathbf{k}} \\\ {\mathbf{v}=8 \mathbf{i}-4 \mathbf{j}+8 \mathbf{k}}\end{array}$$

Using the Midpoint Formula in Space In Exercises \(45-52\) , find the midpoint of the line segment joining the points. $$(0,-2,5),(4,2,7)$$

Using the Midpoint Formula in Space In Exercises \(45-52\) , find the midpoint of the line segment joining the points. $$(-2,8,10),(7,-4,2)$$

Finding the Area of a Triangle In Exercises \(49-52\) , find the area of the triangle with the given vertices. $$(2,3,-5),(-2,-2,0),(3,0,6)$$