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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

```
\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

```
\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

Finding distance using vector equation of a line

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1.50mins

Finding distance using vector equation of a line

Finding Parametric equation of a line converting it to Catesian RectangularForm

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1.39mins

Finding Parametric equation of a line converting it to Catesian RectangularForm

Example of Vector Form of a Line

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1.13mins

Example of Vector Form of a Line

ex6 Finding Point of Intersection in 2D

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2.24mins

ex6 Finding Point of Intersection in 2D

ex7 Finding Point of Intersection in 3D

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3.53mins

ex7 Finding Point of Intersection in 3D

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

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3.02mins

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

Symmetric form of a line

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2.27mins

Symmetric form of a line

Shortest Distance formula

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3.55mins

Shortest Distance formula

Shortest distance from a point to 3D line

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3.10mins

Shortest distance from a point to 3D line

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**

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2.44mins

Shortest Distance to line from a Point in 3D