Determine the coordinates of the point of interse3ction of the line defined by the parametric equations and the plane defined by the scalar equation.

\displaystyle \ell:\left\{\begin{array}{l}x=4+t \\ y=2-2 t \\ z=6+3 t\end{array}\right.

\displaystyle \pi: x+5 y+z-8=0

Verify that the plane and line are parallel, and then determine if they are distinct or coincident.

\displaystyle \begin{array}{llllllll} &3x + 6y + z -5 = 0 \\ &(x,y, z) = (1, 2, -8) +t(2, -1, -1) \end{array}

Solve each linear system in two-space.

\displaystyle \begin{array}{llllllll} &[x, y] = [-12, -7] + s[8, -5]\\ &[x, y] = [2, -1] + t[3, -2] \end{array}

Verify that the plane and line are parallel, and then determine if they are distinct or coincident.

\displaystyle \begin{array}{llllllll} &x + 2y - 5z + 4 = 0 \\ &(x,y, z) = (10, 3, 4) +t(1, 2, 1) \end{array}

Determine if the plane and line intersect. If so, state the solution.

\displaystyle \begin{array}{llllllll} &3x - y + 4z - 8= 0 &(x,y, z) = (3, 0 , 5) +t(7, -11, -8) \end{array}

Determine if the plane and line intersect. If so, state the solution.

\displaystyle \begin{array}{llllllll} &-2x + 6y + 4z - 4= 0\\ &(x,y, z) = (5, -1, 4) +t(1, -2, 3) \end{array}

Determine the number of solutions for each system without solving.

\displaystyle [x, y]=[1,6]+s[3,-2]

\displaystyle [x, y]=[4,4]+t[-6,4]

Determine the number of solutions for each system without solving.

\displaystyle [x, y]=[-17,-7]+t[8,-3]

\displaystyle 8 x-3 y=11

Determine the number of solutions for each system without solving.

\displaystyle -6 x+45 y=33

\displaystyle 10 x-75 y=-55

Determine the number of solutions for each system without solving.

\displaystyle [x, y]=[11,12]+s[2,7]

\displaystyle [x, y]=[2,3]+t[1,4]

Determine if the parallel lines in each pair are distinct or coincident.

\displaystyle [x, y, z]=[5,1,3]+s[2,1,7]

\displaystyle [x, y, z]=[2,3,9]+t[2,1,7]

Determine if the parallel lines in each pair are distinct or coincident.

\displaystyle [x, y, z]=[4,1,0]+s[3,-5,6]

\displaystyle [x, y, z]=[13,-14,18]+t[-3,5,-6]

Determine if the lines in each pair intersect. If so, find the coordinates of the point of intersection.

\displaystyle [x, y, z]=[6,5,-14]+s[-1,1,3]

\displaystyle [x, y, z]=[11,0,-17]+t[4,-1,-6]

Determine if the non-parallel lines in each pair are skew.

\displaystyle [x, y, z]=[4,7,-1]+s[-2,1,2]

\displaystyle [x, y, z]=[1,3,-1]+t[4,-1,2]

The parametric equations of three lines are given. Do these define three different lines, two different lines, or only one line? Explain.

\displaystyle \ell_{1}:\left\{\begin{array}{l}x=2+3 s \\ y=-8+4 s \\ z=1-2 s\end{array}\right. \displaystyle \ell_{2}:\left\{\begin{array}{l}x=4+9 s \\ y=-16+12 s \\ z=2-6 s\end{array}\right. \displaystyle \ell_{3}:\left\{\begin{array}{l}x=3+9 s \\ y=7+12 s \\ z=2+6 s\end{array}\right.

Determine the distance between the skew lines in each pair.

\displaystyle \begin{aligned} \ell_{1}:[x, y, z] &=[3,1,0]+s[1,8,2] \\ \ell_{2}:[x, y, z] &=[-4,2,1]+t[-1,-2,1] \end{aligned}

Determine the distance between the skew lines in each pair.

\displaystyle \begin{aligned} \ell_{1}:[x, y, z] &=[5,2,-3]+s[5,5,1] \\ \ell_{2}:[x, y, z] &=[-1,-4,-4]+t[7,-2,-2] \end{aligned}

These equations represent the sides of a triangle.

\displaystyle \begin{aligned} \ell_{1}:[x, y] &=[-1,-1]+r[5,-1] \\ \ell_{2}:[x, y] &=[7,-10]+s[3,-8] \\ \ell_{3}:[x, y] &=[3,13]+t[2,7] \end{aligned}

a) Determine the intersection of each pair of lines.

b) Find the perimeter of the triangle.

Write equations of two non-parallel lines in three-space that intersect at each point. \displaystyle (1,-7,1)

Write equations of two non-parallel lines in three-space that intersect at each point.

\displaystyle (2,4,-3)

A median of a triangle is the line from a vertex to the midpoint of the opposite side. The point of intersection of the medians of a triangle is called the centroid.

a) The vertices of a triangle are at \displaystyle \mathrm{A}(2,6) \displaystyle \mathrm{B}(10,9) , and \displaystyle \mathrm{C}(9,3) . Find the centroid of the triangle.

b) Plot the points to confirm your answer to part a).

c) Find the centroid of a triangle with vertices at \displaystyle \mathrm{D}(2,-6,8), \mathrm{E}(9,0,2) , and \displaystyle \mathrm{F}(-1,3,-2)

Develop a formula for the solution to each system of equations.

\displaystyle \begin{aligned} a x+b y &=c \\ & d x+e y=f \end{aligned}

Develop a formula for the solution to each system of equations.

\displaystyle [x, y]=[a, b]+s[c, d]

\displaystyle [x, y]=[e, f]+t[g, b]

Math Contest The two overlapping rectangles shown have the same width, but different lengths. Determine the length of the second rectangle.

Math Contest The angle between the planes \displaystyle x+y+2 z=11 and \displaystyle 2 x-y+k z=99 is \displaystyle 60^{\circ} . Determine all possible values of \displaystyle k .