LemonmathSelf-study sheet

Grade 12 · Lesson 5.2

The equation of a line in space

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Goals

Write parametric and Cartesian equations of a line from a point and a direction vector, or from two points.
Decide whether two lines are parallel, coincident, intersecting or skew.
Decide how a line meets a plane; find the point of intersection.
Find the orthogonal projection of a point on a plane and on a line.

Remember

The line through $M_0(x_0, y_0, z_0)$ with direction vector $\vec{u}=(a, b, c)$ : $x=x_0+at$ , $y=y_0+bt$ , $z=z_0+ct$ .
When $abc\neq0$ : $\dfrac{x-x_0}{a}=\dfrac{y-y_0}{b}=\dfrac{z-z_0}{c}$ .
Through two points $A$ , $B$ : $\vec{u}=\overrightarrow{AB}$ . Perpendicular to $(P)$ : $\vec{u}=\vec{n}_P$ .
Two lines: if $\vec{u}_1$ , $\vec{u}_2$ are parallel, the lines are parallel or coincide; if not, solve for $t$ , $s$ : a solution means they intersect, no solution means they are skew.
A line and a plane: substituting gives an equation in $t$ ; one solution means they meet, no solution means parallel, every $t$ means the line lies in the plane.
Projection on $(P)$ : where $(P)$ meets the line through $M$ with direction $\vec{n}$ . Projection on $d$ : $H\in d$ with $\overrightarrow{MH}\cdot\vec{u}=0$ .

Exercises

Exercise 1

Write parametric equations of the line through $A(2, -1, 3)$ with direction vector $\vec{u}=(1, 4, -2)$ . Find the point for $t=-1$ and check whether $B(4, 7, -1)$ lies on the line.

Exercise 2

Write parametric and Cartesian equations of the line through $A(1, 0, -2)$ and $B(3, 2, -1)$ .

Exercise 3

Write the equation of the line $d$ through $M(1, -1, 2)$ perpendicular to $(P): 2x-y+3z+5=0$ . Find the point where $d$ meets $(P)$ .

Exercise 4

Write parametric and Cartesian equations of the line through $M(0, 1, -1)$ parallel to $\Delta: \dfrac{x-1}{2}=\dfrac{y+3}{-1}=\dfrac{z}{3}$ .

Exercise 5

Let $d_1: \begin{cases}x=1+2t\\ y=-t\\ z=3+t\end{cases}$ and $d_2: \dfrac{x-3}{-4}=\dfrac{y+1}{2}=\dfrac{z-4}{-2}$ . Is each claim true or false?

a) $\vec{u}_1=(2, -1, 1)$ is a direction vector of $d_1$ .

b) The point $M(3, -1, 4)$ lies on $d_1$ .

c) $d_1$ is parallel to $d_2$ .

d) $d_1$ is perpendicular to the plane $(P): 2x-y+z=0$ .

Exercise 6

An aeroplane takes off from the point $O(0, 0, 0)$ and flies in a straight line, moving by the vector $\vec{v}=(80, 60, 10)$ every second (unit: metre). When it reaches a height of $1500$ m, how many kilometres from $O$ is the point on the ground directly below it?

Exercise 7

Let $d: x=2+t$ , $y=1-t$ , $z=3+2t$ . Decide the position of $d$ relative to $(P_1): x+y-1=0$ , $(P_2): x+3y+z-8=0$ , $(P_3): x-y+z-6=0$ .

Exercise 8

Decide the position of the lines $d_1: x=1+t$ , $y=1-t$ , $z=2t$ and $d_2: x=1+2s$ , $y=-2+s$ , $z=5-s$ . If they intersect, find the point of intersection.

Exercise 9

(Revisiting Operations on vectors in coordinates.) In the previous lesson we proved that $A(1, 2, -1)$ , $B(3, 1, 1)$ , $C(7, -1, 5)$ are collinear using parallel vectors. Write parametric equations of the line $AB$ and give the value of $t$ for the point $C$ .

Exercise 10 Challenge

A laser beam starts at $A(1, 2, 3)$ , reflects off a flat mirror lying in the plane $(P): x+2y+2z-2=0$ , and then reaches $B(3, 2, 2)$ . Find the point $M$ of the mirror where it reflects. (Hint: the path $A\to M\to B$ is shortest when $M$ lies on the segment $A'B$ , where $A'$ is the reflection of $A$ in $(P)$ .)

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. In the system

Oxyz

, a direction vector of the line

d: \begin{cases}x=1-2t\\ y=3\\ z=-1+t\end{cases}

(1, 3, -1)

.B.

(-2, 3, 1)

.C.

(-2, 0, 1)

.D.

(2, 0, 1)

Question 2. Which point lies on the line

\dfrac{x-1}{2}=\dfrac{y+2}{-1}=\dfrac{z-3}{3}

(-1, 2, -3)

.B.

(2, -1, 3)

.C.

(3, -1, 6)

.D.

(1, -2, 3)

Question 3. The line through

A(1, 0, 2)

perpendicular to

(P): 2x-y+z-3=0

has equations

x=1+2t

y=-t

z=2+t

.B.

x=2+t

y=-1

z=1+2t

.C.

x=1+2t

y=t

z=2+t

.D.

x=1+2t

y=-t

z=2-3t

Question 4. Let

d_1: x=1+t

y=2t

z=3-t

and

d_2: x=s

y=1+s

z=2s

. The lines

d_1

and

d_2

are

A. parallel.B. intersecting.C. skew.D. coincident.

Part II. True or false

Question 1. Let

d: x=1+t

y=2-t

z=-1+2t

and

(P): x+y+z-4=0

a) $\vec{u}=(1, -1, 2)$ is a direction vector of $d$ .	T / F
b) $d$ is parallel to $(P)$ .	T / F
c) $d$ meets $(P)$ at the point $(2, 1, 1)$ .	T / F
d) $d$ is perpendicular to the plane $(Q): x-y+2z+7=0$ .	T / F

Part III. Short answer

Question 1. The line

d: x=2+t

y=-1+2t

z=3-t

meets the plane

(P): x+y+z-10=0

M(a, b, c)

. Find

a+b+c

Question 2. Find the distance from

M(2, 3, 1)

to the line

d: x=1+2t

y=-1+t

z=1+2t

(to 2 decimal places).

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $x=x_0+at$ , $y=y_0+bt$ , $z=z_0+ct$ .

Steps:

Write the parametric equations.
Substitute $t=-1$ .
Find $t$ from the x-coordinate of $B$ , then check the other two coordinates.

Exercise 2.

What you need: $\vec{u}=\overrightarrow{AB}$ .

Steps:

Find $\overrightarrow{AB}$ .
Write both forms using the point $A$ .

Exercise 3.

What you need: if $d\perp(P)$ then $\vec{u}_d=\vec{n}_P$ ; for the intersection, substitute the parametric equations into $(P)$ .

Steps:

Write $d$ with $\vec{u}=(2, -1, 3)$ .
Substitute into $(P)$ and solve for $t$ .

Exercise 4.

What you need: parallel lines share a direction vector; read $\vec{u}$ from the denominators.

Steps:

Read off $\vec{u}=(2, -1, 3)$ .
Write the equations through $M$ .

Exercise 5.

What you need: if the direction vectors are parallel, the lines are parallel or coincide; check one more point.

Steps:

Read off $\vec{u}_1$ , $\vec{u}_2$ .
Substitute $M$ into $d_1$ (find $t$ ).
Compare $\vec{u}_1$ with $\vec{n}_P$ .

Exercise 6.

What you need: the position after $t$ seconds is $(80t, 60t, 10t)$ .

Steps:

Solve $10t=1500$ .
The point on the ground is $(80t, 60t, 0)$ ; find its distance from $O$ .

Exercise 7.

What you need: substitute the parametric equations into the equation of the plane.

Steps:

Substitute into each plane and simplify in $t$ .
Count the solutions.

Exercise 8.

What you need: if the direction vectors are not parallel, solve the system in $t$ , $s$ .

Steps:

Compare $(1, -1, 2)$ and $(2, 1, -1)$ .
Solve the first two equations, then test the third.

Exercise 9.

What you need: $\vec{u}=\overrightarrow{AB}$ ; $C$ lies on $AB$ when one value of $t$ works for all three coordinates.

Steps:

$\overrightarrow{AB}=(2, -1, 2)$ .
Solve $1+2t=7$ and check.

Exercise 10.

What you need: the projection $H$ of $A$ on $(P)$ ; $H$ is the midpoint of $AA'$ .

Steps:

Find $H$ , then $A'=2H-A$ (in coordinates).
Use the line $A'B$ and find where it meets $(P)$ .

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