3.4 The Intersection of Two Lines
Chapter
Chapter 3
Section
3.4
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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?

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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} 

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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} 

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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} 

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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 

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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 

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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 

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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} 

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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} 

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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} 

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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} 

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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} 

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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.
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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.
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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.

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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.

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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.

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2.53mins
Q8

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

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1.03mins
Q9

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

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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.

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0.00mins
Q11

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?

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2.05mins
Q12
Lectures 12 Videos