Direction vector of the line through A and B?

\overrightarrow{AB}=\mathbf b-\mathbf a (end minus start)

Vector equation of the line through A and B?

\mathbf r=\mathbf a+\lambda(\mathbf b-\mathbf a)

Cartesian form of a line in 3D?

\frac{x-a1}{d1}=\frac{y-a2}{d2}=\frac{z-a3}{d3}

Cartesian form of \mathbf r=(1,0,2)+\lambda(3,4,5)?

\frac{x-1}{3}=\frac{y}{4}=\frac{z-2}{5}

Point and direction from \frac{x-2}{3}=\frac{y+1}{-2}=\frac{z-4}{5}?

Point (2,-1,4); direction (3,-2,5)

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

Find \lambda from one component, then check the other two give the same \lambda.

When are two lines parallel?

When their direction vectors are multiples of each other.

Two parallel lines: how do you tell if they are the same line?

Test whether a point on one line lies on the other.

Cartesian form when the direction is (0,2,5) through (3,1,-2)?

x=3,\;\frac{y-1}{2}=\frac{z+2}{5}

Does the base point or direction vector in a line equation have to be unique?

No. Any point on the line and any non-zero multiple of the direction can be used.

What does \lambda=0 give in \mathbf r=\mathbf a+\lambda(\mathbf b-\mathbf a)? And \lambda=1?

\lambda=0 gives A; \lambda=1 gives B.

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

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Vector equation of a line through $\mathbf a$ with direction $\mathbf d$?

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Vector equation of a line through $\mathbf a$ with direction $\mathbf d$ ?: $\mathbf r=\mathbf a+\lambda\mathbf d$
Direction vector of the line through $A$ and $B$ ?: $\overrightarrow{AB}=\mathbf b-\mathbf a$ (end minus start)
Vector equation of the line through $A$ and $B$ ?: $\mathbf r=\mathbf a+\lambda(\mathbf b-\mathbf a)$
Cartesian form of a line in 3D?: $\frac{x-a_1}{d_1}=\frac{y-a_2}{d_2}=\frac{z-a_3}{d_3}$
Cartesian form of $\mathbf r=(1,0,2)+\lambda(3,4,5)$ ?: $\frac{x-1}{3}=\frac{y}{4}=\frac{z-2}{5}$
Point and direction from $\frac{x-2}{3}=\frac{y+1}{-2}=\frac{z-4}{5}$ ?: Point $(2,-1,4)$ ; direction $(3,-2,5)$
How do you test whether a point lies on a line?: Find $\lambda$ from one component, then check the other two give the same $\lambda$ .
When are two lines parallel?: When their direction vectors are multiples of each other.
Two parallel lines: how do you tell if they are the same line?: Test whether a point on one line lies on the other.
Cartesian form when the direction is $(0,2,5)$ through $(3,1,-2)$ ?: $x=3,\;\frac{y-1}{2}=\frac{z+2}{5}$
Does the base point or direction vector in a line equation have to be unique?: No. Any point on the line and any non-zero multiple of the direction can be used.
What does $\lambda=0$ give in $\mathbf r=\mathbf a+\lambda(\mathbf b-\mathbf a)$ ? And $\lambda=1$ ?: $\lambda=0$ gives $A$ ; $\lambda=1$ gives $B$ .

Exam questions on Vector and Cartesian equations of lines

The line $l$ passes through the points $A(1,2,-1)$ and $B(3,-2,3)$ .
Show that the point $C(5,-6,7)$ lies on $l$ .2 marks
A line $l_1$ has Cartesian equation $\frac{x-2}{3}=\frac{y+1}{-2}=\frac{z-4}{5}$ .
Find the coordinates of the point on $l_1$ at which $z=-1$ .2 marks
The line $l_2$ has vector equation $\mathbf r=\begin{pmatrix} 4 \\ -3 \\ 1 \end{pmatrix}+\mu\begin{pmatrix} 2 \\ -1 \\ 3 \end{pmatrix}$ .
Find a Cartesian equation of $l_2$ .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).