Vector equations of lines and curves

Write lines in vector form

\mathbf{r} = \mathbf{a}+\lambda\mathbf{b}

and parametric form; convert between forms; find points on a line; determine whether two lines are parallel, intersecting or skew; find intersection points and angles between lines; model simple curves parametrically.

Worked examples

Reading off a point and converting to parametric form

Straightforward

Problem

A line has equation

\mathbf{r} = (2,-1,3)+\lambda(1,2,-1)

. Find the point where

\lambda = 3

, and write parametric equations.

Intersection of two lines

Moderate

Problem

Determine whether

L_1: \mathbf{r} = (1,2,3)+\lambda(2,0,1)

and

L_2: \mathbf{r} = (3,2,4)+\mu(-1,0,2)

intersect, and if so find the point.

Identifying skew lines

Challenging

Problem

Show that

L_1: \mathbf{r} = (0,0,0)+\lambda(1,0,0)

and

L_2: \mathbf{r} = (0,1,1)+\mu(0,1,0)

are skew.

Practise

Q1·Straightforward

A line has equation

\mathbf{r} = \mathbf{i}+2\mathbf{j}-\mathbf{k}+\lambda(3\mathbf{i}-\mathbf{j}+2\mathbf{k})

. Find the

x

-coordinate of the point where

\lambda = 2

Q2·Straightforward

A line has equation

\mathbf{r} = (0,-3,1)+t(1,2,-1)

. Find the value of

t

when the

z

-coordinate equals

-2

Q3·Straightforward

Find the distance

|AB|

where

A = (2,1,3)

and

B = (5,-3,3)

Q4·Straightforward

The point

P(5,-5,4)

lies on the line

\mathbf{r} = (1,1,2)+\lambda(-2,3,-1)

. Find the value of

\lambda

Q5·Moderate

A line passes through

A(1,-1,3)

with direction vector

(1,1,-1)

. Find the

z

-coordinate of the point on the line where

x = 7

Q6·Moderate

Line

L_1

has direction vector

(2,-1,4)

and line

L_2

has direction vector

(6,-3,12)

. Find the scalar

k

such that

\mathbf{d}_{L_2} = k\,\mathbf{d}_{L_1}

Q7·Moderate

Find the acute angle, in degrees, between lines with direction vectors

(1,2,2)

and

(2,1,-2)

Q8·Moderate

Lines

L_1: \mathbf{r} = (1,2,3)+\lambda(2,0,1)

and

L_2: \mathbf{r} = (3,2,4)+\mu(-1,0,2)

intersect at a point

P

. Find the

z

-coordinate of

P

Q9·Moderate

Find the acute angle, in degrees, between two lines whose direction vectors are

(1,1,0)

and

(0,1,1)

Q10·Challenging

Consider lines

L_1: \mathbf{r} = (1,0,0)+\lambda(0,1,0)

and

L_2: \mathbf{r} = (0,0,1)+\mu(1,0,0)

. Which best describes their relationship?

Skew — they are not parallel, and setting parametric equations equal gives no solutionIntersecting — they meet where

\lambda = 0

and

\mu = 1

Parallel — the direction vectors are proportionalIdentical — both pass through the origin

Q11·Challenging

A curve is modelled parametrically by

\mathbf{r}(t) = (\cos t,\,\sin t,\,t/\pi)

for

0 \le t \le 2\pi

. Find the

z

-coordinate of the point where

x = 1

and

y = 0

for the second time (i.e. when

t > 0

Q12·Challenging

Find the acute angle, in degrees, between two lines whose direction vectors are

(2,3,6)

and

(2,-2,1)

. Round to two decimal places.

Open Math

Vector equations of lines and curves

Vectors · MEX-12-05

Name:

Date:

Q1Straightforward

A line has equation

\mathbf{r} = \mathbf{i}+2\mathbf{j}-\mathbf{k}+\lambda(3\mathbf{i}-\mathbf{j}+2\mathbf{k})

. Find the

x

-coordinate of the point where

\lambda = 2

Q2Straightforward

A line has equation

\mathbf{r} = (0,-3,1)+t(1,2,-1)

. Find the value of

t

when the

z

-coordinate equals

-2

Q3Straightforward

Find the distance

|AB|

where

A = (2,1,3)

and

B = (5,-3,3)

Q4Straightforward

The point

P(5,-5,4)

lies on the line

\mathbf{r} = (1,1,2)+\lambda(-2,3,-1)

. Find the value of

\lambda

Q5Moderate

A line passes through

A(1,-1,3)

with direction vector

(1,1,-1)

. Find the

z

-coordinate of the point on the line where

x = 7

Q6Moderate

Line

L_1

has direction vector

(2,-1,4)

and line

L_2

has direction vector

(6,-3,12)

. Find the scalar

k

such that

\mathbf{d}_{L_2} = k\,\mathbf{d}_{L_1}

Q7Moderate

Find the acute angle, in degrees, between lines with direction vectors

(1,2,2)

and

(2,1,-2)

Q8Moderate

Lines

L_1: \mathbf{r} = (1,2,3)+\lambda(2,0,1)

and

L_2: \mathbf{r} = (3,2,4)+\mu(-1,0,2)

intersect at a point

P

. Find the

z

-coordinate of

P

Q9Moderate

Find the acute angle, in degrees, between two lines whose direction vectors are

(1,1,0)

and

(0,1,1)

Q10Challenging

Consider lines

L_1: \mathbf{r} = (1,0,0)+\lambda(0,1,0)

and

L_2: \mathbf{r} = (0,0,1)+\mu(1,0,0)

. Which best describes their relationship?

A.Skew — they are not parallel, and setting parametric equations equal gives no solution
B.Intersecting — they meet where $\lambda = 0$ and $\mu = 1$
C.Parallel — the direction vectors are proportional
D.Identical — both pass through the origin

Q11Challenging

A curve is modelled parametrically by

\mathbf{r}(t) = (\cos t,\,\sin t,\,t/\pi)

for

0 \le t \le 2\pi

. Find the

z

-coordinate of the point where

x = 1

and

y = 0

for the second time (i.e. when

t > 0

Q12Challenging

Find the acute angle, in degrees, between two lines whose direction vectors are

(2,3,6)

and

(2,-2,1)

. Round to two decimal places.

Worked solutions and answers at openmath.au/year-12/extension-2/further-work-with-vectors/vector-equations-of-lines-and-curves