Textbook

Calculus and Vectors Nelson
Chapter

Chapter 9
Section

Relations between Points, Lines and Planes Test

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Solutions
7 Videos

a) Determine the point of intersection for the lines having equations `\vec{r} = (4, 2, 6) +s(1, ,3 11), x \in \mathbb{R}`

and `\vec{r} = (5, -1, 4) +t(2, 0, 9), t\in \mathbb{R}`

.

b) Verify that the intersection point of these two lines is on the plane.

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Q1

a) Determine the distance from point `A(3, 2, 3)`

to `\pi: 8x -8y + 4z - 7 = 0`

.

b) Determine the distance between the planes `\pi_1: 2x - y -2z - 16 = 0`

and `\pi_2: 2x - y + 2z + 24 = 0`

.

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Q2

a) Determine the equation of the line of intersection `L`

between the planes `\pi_1: 2x + 3y - z =3`

and `\pi_2: -x + y + z = 1`

.

b) Determine the point of intersection between `L`

and xz-plane.

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Q3

Solve the following system of equations:

```
\displaystyle
\begin{array}{lllll}
& \phantom{.} x - y + z = 10 \\
& \phantom{.} 2x + 3y -2z = -21 \\
& \phantom{.} \frac{1}{2}x + \frac{2}{5}y + \frac{1}{4}z = -\frac{1}{2}
\end{array}
```

Explain what your solution means geometrically.

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Q4

Solve the following system of equations:

```
\displaystyle
\begin{array}{lllll}
& \phantom{.} x - y + z = -1 \\
& \phantom{.} 2x + 2y -z = 0 \\
& \phantom{.} x -5y +4z = -3
\end{array}
```

Explain what your solution means geometrically.

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Q5

The three planes `x + y + z = 0, x + 2y + 2z = 1`

, and `2x - y + mz = n`

intersect in a line.

a) Determine the values of `m`

and `n`

for which this is true.

b) What is the equation of the line?

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Q6

Determine the distance between the skew lines with equations

```
\displaystyle
\begin{array}{lllll}
& \phantom{.} L_1: & \vec{r} = (-1, -3, 0) +s(1, 1, 1), s \in \mathbb{R}\\
& \phantom{.} L_2: & \vec{r} = (-5, 5, -8) +t(1, 2, 5), t \in \mathbb{R}
\end{array}
```

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Q7