8.4 Intersections of Lines in Two-Space and Three-Space
Chapter
Chapter 8
Section
8.4
Solutions 25 Videos

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

Q1

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} 

Q2a

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} 

Q2c

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} 

Q2d

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} 

Q3a

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} 

Q3b

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]

0.13mins
Q3c

Determine the number of solutions for each system without solving.

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

\displaystyle 8 x-3 y=11

0.29mins
Q3d

Determine the number of solutions for each system without solving.

\displaystyle -6 x+45 y=33

\displaystyle 10 x-75 y=-55

Q3e

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]

0.14mins
Q3f

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]

Q4a

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]

Q4b

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]

Q5a

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]

Q6a

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.

Q8

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} 

Q9a

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}

Q9d

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.

Q10

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

2.45mins
Q16a

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

\displaystyle (2,4,-3)

Q16b

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)

Q17

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} 

Q18a

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 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 .