3.4 The Intersection of Two Lines
Chapter
Chapter 3
Section
3.4
Lectures 12 Videos

Finding distance using vector equation of a line

1.50mins
Finding distance using vector equation of a line

Finding Parametric equation of a line converting it to Catesian RectangularForm

1.39mins
Finding Parametric equation of a line converting it to Catesian RectangularForm

ex8 Finding intersection of two lines in 3D with no solution

3.02mins
ex8 Finding intersection of two lines in 3D with no solution

Shortest Distance to line from a Point in 3D

This video has an error at the end with distance calculation - forgot to square root for lengths

2.44mins
Shortest Distance to line from a Point in 3D
Solutions 21 Videos

Line 1 intersects both the x-axis and the y-axis. Line 2 intersects only the z-axis. Neither contains the origin. Must the two lines be parallel or skew, or can they intersect?

1.18mins
Q1

Find the intersection point of the following pairs of lines.

\displaystyle \begin{array}{lllll} & \phantom{.} 2x+ 5y + 15 = 0 \\ & \phantom{.} 3x -4y + 11 = 0 \end{array} 

0.59mins
Q2a

Find the intersection point of the following pairs of lines.

\displaystyle \begin{array}{lllll} & \phantom{.} \vec{r} = (-3, -6) + s(1, 1) \\ & \phantom{.} \vec{r} =(4, -8) + t(1, 2) \end{array} 

1.45mins
Q2b

Determine whether the following pairs of lines are coincident, parallel and distinct, or neither.

\displaystyle \frac{x-3}{10} = \frac{y - 8}{-4} 

1.14mins
Q3a

Determine whether the following pairs of lines are coincident, parallel and distinct, or neither.

\displaystyle x = 6 -18s, y = 12 + 3s\\ x = 8-6t, y = 4+ 9t 

0.28mins
Q3b

Determine whether the following pairs of lines are coincident, parallel and distinct, or neither.

\displaystyle x =8 + 12s, y = 4-4s, z = 3- 6s \\ x = 2 - 4t, y = 2+ t, z = 6 + 2t 

0.55mins
Q3c

Determine whether the following pairs of lines are coincident, parallel and distinct, or neither.

\displaystyle \frac{x+ 4}{3} = \frac{y - 12}{4} = \frac{z - 3}{6} \\ \frac{x}{\frac{1}{2}} = \frac{y - 10}{\frac{2}{3}} = z + 5 

1.40mins
Q3d

Find the intersection point of each of the following pairs of lines. If they do not meet, determine whether they are parallel and distinct or skew.

\displaystyle \begin{array}{lllll} & \vec{r} = (-2, 0, -3) + t(5, 1, 3) \\ & \vec{r} = (5, 8, -6) + u(-1, 2, -3) \end{array} 

2.50mins
Q4a

Find the intersection point of each of the following pairs of lines. If they do not meet, determine whether they are parallel and distinct or skew.

\displaystyle \begin{array}{lllll} & x = 1 + t, y = 1+ 2t, z 1- 3t \\ & x = 3 - 2u, y = 5- 4u, z = -5 + 6u \end{array} 

1.04mins
Q4b

Find the intersection point of each of the following pairs of lines. If they do not meet, determine whether they are parallel and distinct or skew.

\displaystyle \begin{array}{lllll} & \vec{r} = (2, -1, 0) + t(1, 2, -3) \\ & \vec{r} = (-1, 1, 2) +u(-2, 1, 1) \end{array} 

2.09mins
Q4c

Find the intersection point of each of the following pairs of lines. If they do not meet, determine whether they are parallel and distinct or skew.

\displaystyle \begin{array}{lllll} & \vec{r} = (1 + t, 2 + t, -t) \\ & \vec{r} = (3- 2u, 4 -2u, -1 + 2u) \end{array} 

0.53mins
Q4d

Find the intersection point of each of the following pairs of lines. If they do not meet, determine whether they are parallel and distinct or skew.

\displaystyle \begin{array}{lllll} & \frac{x - 3}{4} = y - 2 = z - 2 \\ & \frac{x - 2}{-3} = \frac{y + 1}{2} = \frac{z - 2}{-1} \end{array} 

1.30mins
Q4e

Consider the lines \vec{r} = (1, -1, 1)+ t( 3, 2, 1) and \vec{r} = (-2, -3, 0) + u(1, 2, 3).

• Find a vector equation for the line perpendicular to both of the given lines that passes through their point of intersection.
2.21mins
Q5a

Consider the lines \vec{r} = (1, -1, 1)+ t( 3, 2, 1) and \vec{r} = (-2, -3, 0) + u(1, 2, 3).

• Find a vector equation for the line perpendicular to both of the given lines that passes through their point of intersection.
0.58mins
Q5b

Show that the lines \vec{r} = (4, 7, -1) + t(4, 8, -4) and \vec{r} = (1, 5, 4)+ u(-1, 2, 3) intersect at right angles and find the point of intersection.

1.09mins
Q6

If they exist, find the x, y, and z intercepts of the line x = 24+ 7t, y = 4+ t, z = -20 -5t.

1.21mins
Q7

Find the point at which the normal through the point (3, -4) to the line 10x + 4y - 101 = 0 intersects the line.

2.53mins
Q8

What are the possible ways that three lines in a plane can intersect? Describe them all with diagrams.

1.03mins
Q9

What are the possible ways that three lines in space can intersect? Describe them all with diagrams.

0.40mins
Q10

Find the equation of the line through the point (-5, -4, 2) that intersects the line at \vec{r} = (7, -13, 8) + t(1, 2, -2) at 90°. Determine the point of intersection.

Find the points of intersection of the line \vec{r} = (0, 5, 3) + t(I, -3, -2) with the sphere x^2 + y^2 + z^2 = 6. Is the segment of the line between the intersection points a diameter of the sphere?