Vector equation of a line through C and D?

\mathbf{r}=\mathbf{c}+t(\mathbf{d}-\mathbf{c})

What does t represent?

A scalar parameter; each value gives one point on the line.

How do you test whether a point lies on a line?

Find t from one component, then check the other two components agree.

When are two lines parallel?

When their direction vectors are scalar multiples of each other.

How do you find where two lines meet?

Equate components using different parameters, solve two equations, check the third, then substitute back.

Why use different parameters s and t for two lines?

The lines reach the common point at different parameter values.

Lines in 3D that are not parallel and do not intersect.

What shows two lines are skew?

Non-parallel directions and inconsistent component equations.

Can parallel lines be skew?

No. Skew lines are never parallel.

Where does a line meet the plane z=0?

Set the \mathbf{k} component to 0, solve for t and substitute.

Is the equation of a line unique?

No: any point on the line and any non-zero multiple of the direction vector will do.

Vector equations of linesEdexcel International A Level Maths: Flashcards

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Vector equation of a line through $A$ parallel to $\mathbf{b}$?

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Vector equation of a line through $A$ parallel to $\mathbf{b}$ ?: $\mathbf{r}=\mathbf{a}+t\mathbf{b}$
Vector equation of a line through $C$ and $D$ ?: $\mathbf{r}=\mathbf{c}+t(\mathbf{d}-\mathbf{c})$
What does $t$ represent?: A scalar parameter; each value gives one point on the line.
How do you test whether a point lies on a line?: Find $t$ from one component, then check the other two components agree.
When are two lines parallel?: When their direction vectors are scalar multiples of each other.
How do you find where two lines meet?: Equate components using different parameters, solve two equations, check the third, then substitute back.
Why use different parameters $s$ and $t$ for two lines?: The lines reach the common point at different parameter values.
What are skew lines?: Lines in 3D that are not parallel and do not intersect.
What shows two lines are skew?: Non-parallel directions and inconsistent component equations.
Can parallel lines be skew?: No. Skew lines are never parallel.
Where does a line meet the plane $z=0$ ?: Set the $\mathbf{k}$ component to 0, solve for $t$ and substitute.
Is the equation of a line unique?: No: any point on the line and any non-zero multiple of the direction vector will do.

Exam questions on Vector equations of lines

The line $l_1$ passes through the points $A(2,-1,3)$ and $B(5,1,1)$ .
Find the coordinates of the point where $l_1$ crosses the plane $z=0$ .2 marks
The line $m$ passes through the points $C(1,4,-2)$ and $D(3,3,0)$ .
Determine whether the point $E(-3,6,-6)$ lies on $m$ .2 marks
Line $l_1$ has equation $\mathbf{r}=(\mathbf{i}+\mathbf{j}+2\mathbf{k})+s(2\mathbf{i}+\mathbf{j}-\mathbf{k})$ and line $l_2$ has equation $\mathbf{r}=(\mathbf{i}+4\mathbf{j}+5\mathbf{k})+t(\mathbf{i}-\mathbf{j}-2\mathbf{k})$ , where $s$ and $t$ are scalar parameters.
Show that $l_1$ and $l_2$ intersect, and find the coordinates of the point of intersection.3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).