Direction of the line through A and B?

The position vector of B minus that of A, \overrightarrow{AB}.

Cartesian form of a line in 3D?

\frac{x-a1}{b1}=\frac{y-a2}{b2}=\frac{z-a3}{b3}.

Cartesian form of \mathbf r=(2,-1,4)+\lambda(3,2,-4)?

\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-4}{-4}.

Convert a Cartesian line to vector form?

Set each fraction equal to \lambda; the numerators give the point, the denominators the direction.

How do you write a line whose direction has a zero component?

That coordinate is constant, e.g. y=a2, with the other two fractions equal.

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

Find \lambda from one coordinate, then check the other two.

How do you find where two lines meet?

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

When are two lines parallel?

Their direction vectors are scalar multiples of each other.

When are two lines skew?

They are not parallel and do not intersect.

Why use a different parameter for each line?

The lines reach the intersection at different parameter values.

Vector and Cartesian equations of a lineEdexcel A-Level Further Maths: Flashcards

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Vector equation of a line?

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Vector equation of a line?: $\mathbf r=\mathbf a+\lambda\mathbf b$ : $\mathbf a$ a point, $\mathbf b$ the direction.
Direction of the line through $A$ and $B$ ?: The position vector of $B$ minus that of $A$ , $\overrightarrow{AB}$ .
Cartesian form of a line in 3D?: $\frac{x-a_1}{b_1}=\frac{y-a_2}{b_2}=\frac{z-a_3}{b_3}$ .
Cartesian form of $\mathbf r=(2,-1,4)+\lambda(3,2,-4)$ ?: $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-4}{-4}$ .
Convert a Cartesian line to vector form?: Set each fraction equal to $\lambda$ ; the numerators give the point, the denominators the direction.
How do you write a line whose direction has a zero component?: That coordinate is constant, e.g. $y=a_2$ , with the other two fractions equal.
How do you test if a point lies on a line?: Find $\lambda$ from one coordinate, then check the other two.
How do you find where two lines meet?: Equate components using different parameters, solve two equations, check the third.
What does it mean if the third equation fails?: The lines do not intersect.
When are two lines parallel?: Their direction vectors are scalar multiples of each other.
When are two lines skew?: They are not parallel and do not intersect.
Can non-parallel lines in 3D fail to meet?: Yes: they are then skew.
Why use a different parameter for each line?: The lines reach the intersection at different parameter values.

Exam questions on Vector and Cartesian equations of a line

The line $l_1$ passes through the points $A(2,-1,4)$ and $B(5,1,0)$ .
Determine whether the point $C(11,5,-12)$ lies on $l_1$ .2 marks
The lines $l_2$ and $l_3$ have vector equations $\mathbf r=\begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix}+\mu\begin{pmatrix} 2 \\ -1 \\ 1 \end{pmatrix}$ and $\mathbf r=\begin{pmatrix} 5 \\ 0 \\ 4 \end{pmatrix}+t\begin{pmatrix} -4 \\ 2 \\ -2 \end{pmatrix}$ .
Explain why $l_2$ and $l_3$ do not intersect.2 marks
Three lines are given by $l_4$ : $\mathbf r=\begin{pmatrix} 3 \\ 1 \\ -2 \end{pmatrix}+\lambda\begin{pmatrix} 1 \\ 2 \\ -1 \end{pmatrix}$ , $l_5$ : $\mathbf r=\begin{pmatrix} 3 \\ 6 \\ -7 \end{pmatrix}+\mu\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ and $l_6$ : $\mathbf r=\begin{pmatrix} 1 \\ 4 \\ 0 \end{pmatrix}+\nu\begin{pmatrix} 0 \\ 1 \\ 1 \end{pmatrix}$ .
Show that $l_4$ and $l_5$ intersect and find the position vector of their 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).