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Further work with vectors

Extend two-dimensional vector work into three dimensions: component form using

\mathbf{i}

,

\mathbf{j}

,

\mathbf{k}

; the dot product and angles in 3D; vector equations of lines and parametric form; intersection and skew lines; geometric and coordinate-geometry proofs using vectors.

Vectors · MEX-12-05

Sub-topics

Three-dimensional vectors

Represent vectors in 3D using components

\mathbf{i}

\mathbf{j}

\mathbf{k}

; find magnitudes and unit vectors; compute the dot product; find angles between vectors in 3D; determine perpendicularity and parallelism.

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.

Geometric proof with vectors

Prove geometric and coordinate-geometry results using vectors in 2D and 3D; prove perpendicularity using the dot product; prove parallelism using scalar multiples; prove results about midpoints, diagonals, medians, and classic theorems such as the angle in a semicircle.

Open Math

Further work with vectors

Vectors · MEX-12-05

Name:

Date:

Three-dimensional vectors

Q1Straightforward

Find

|\mathbf{u}|

where

\mathbf{u} = 2\mathbf{i} + 3\mathbf{j} + 6\mathbf{k}

.

Q2Straightforward

Find

\mathbf{a}\cdot\mathbf{b}

where

\mathbf{a} = \mathbf{i}+2\mathbf{j}+3\mathbf{k}

and

\mathbf{b} = 4\mathbf{i}-\mathbf{j}+2\mathbf{k}

.

Q3Straightforward

Let

\mathbf{a} = 3\mathbf{i}-2\mathbf{j}+\mathbf{k}

and

\mathbf{b} = -\mathbf{i}+4\mathbf{j}+2\mathbf{k}

. Find the

\mathbf{k}

-component of

\mathbf{a}+\mathbf{b}

.

Q4Straightforward

Find

|\mathbf{v}|

where

\mathbf{v} = -6\mathbf{i}+8\mathbf{k}

.

Q5Straightforward

Find

|\mathbf{w}|

where

\mathbf{w} = 5\mathbf{j}-12\mathbf{k}

.

Q6Straightforward

Find

\mathbf{u}\cdot\mathbf{v}

where

\mathbf{u} = \mathbf{i}-\mathbf{k}

and

\mathbf{v} = 2\mathbf{i}+3\mathbf{j}+4\mathbf{k}

.

Q7Moderate

Find the angle, in degrees, between

\mathbf{a} = \mathbf{i}+\mathbf{j}

and

\mathbf{b} = \mathbf{i}+\mathbf{k}

.

Q8Moderate

Find the value of

m

such that

\mathbf{u} = m\mathbf{i}-\mathbf{j}+3\mathbf{k}

is perpendicular to

\mathbf{v} = 2\mathbf{i}+m\mathbf{j}+\mathbf{k}

.

Q9Moderate

Find the angle, in degrees, between

\mathbf{u} = (2,-1,2)

and

\mathbf{v} = (1,2,2)

. Round to two decimal places.

Q10Moderate

Find the

x

-component of the unit vector in the direction of

\mathbf{v} = 6\mathbf{i}+3\mathbf{j}+2\mathbf{k}

. Give your answer as a fraction.

Q11Moderate

Which statement correctly describes

\mathbf{a} = (2,-4,6)

and

\mathbf{b} = (-1,2,-3)

?

A.They are parallel, because $\mathbf{a} = -2\mathbf{b}$
B.They are perpendicular, because $\mathbf{a}\cdot\mathbf{b} = 0$
C.They are neither parallel nor perpendicular
D.They are parallel, because $|\mathbf{a}| = 2|\mathbf{b}|$

Q12Moderate

Let

\mathbf{a} = (1,-2,2)

and

\mathbf{b} = (2,1,-2)

. Find

|\mathbf{a}+\mathbf{b}|

, rounding to two decimal places.

Q13Challenging

Find the value of

t

such that

(t,3,-2)

and

(1,t,4)

are perpendicular.

Q14Challenging

Find the angle, in degrees, that the main diagonal of a unit cube makes with one of its edges. Take the diagonal from

(0,0,0)

to

(1,1,1)

and the edge along the positive

x

-axis. Round to two decimal places.

Q15Challenging

Vectors

\mathbf{a}

and

\mathbf{b}

satisfy

|\mathbf{a}| = 3

,

|\mathbf{b}| = 4

and

\mathbf{a}\cdot\mathbf{b} = 6

. Find the angle between them in degrees.

Vector equations of lines and curves

Q16Straightforward

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

.

Q17Straightforward

A line has equation

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

. Find the value of

t

when the

z

-coordinate equals

-2

.

Q18Straightforward

Find the distance

|AB|

where

A = (2,1,3)

and

B = (5,-3,3)

.

Q19Straightforward

The point

P(5,-5,4)

lies on the line

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

. Find the value of

\lambda

.

Q20Moderate

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

.

Q21Moderate

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}

.

Q22Moderate

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

(1,2,2)

and

(2,1,-2)

.

Q23Moderate

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

.

Q24Moderate

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

(1,1,0)

and

(0,1,1)

.

Q25Challenging

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

Q26Challenging

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

).

Q27Challenging

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.

Geometric proof with vectors

Q28Straightforward

Points

A

,

B

and

C

have position vectors

(1,-2,3)

,

(3,0,5)

and

(5,2,7)

respectively. Find the scalar

t

such that

\overrightarrow{AB} = t\,\overrightarrow{AC}

.

Q29Straightforward

Triangle

ABC

has vertices

A = (1,0,0)

,

B = (0,2,0)

and

C = (0,0,3)

. Find the

z

-coordinate of the centroid

G

.

Q30Straightforward

Position vectors of

A

and

B

are

\mathbf{a} = (2,1,-3)

and

\mathbf{b} = (-4,3,1)

. Find

|\overrightarrow{AB}|

, the distance from

A

to

B

. Round to two decimal places.

Q31Moderate

Let

ABCD

be a quadrilateral with vertices

A

,

B

,

C

,

D

having position vectors

\mathbf{a}

,

\mathbf{b}

,

\mathbf{c}

,

\mathbf{d}

. Let

P

,

Q

,

R

,

S

be the midpoints of

AB

,

BC

,

CD

and

DA

respectively. Prove that

PQRS

is a parallelogram.

Q32Moderate

Let triangle

ABC

have position vectors

\mathbf{a}

,

\mathbf{b}

,

\mathbf{c}

. Let

M

be the midpoint of

BC

and

N

be the midpoint of

AC

. Prove that

MN

is parallel to

AB

and that

|MN| = \tfrac{1}{2}|AB|

.

Q33Moderate

Let

ABCD

be a parallelogram where

\overrightarrow{AB} = \mathbf{b}

and

\overrightarrow{AD} = \mathbf{d}

. Using

A

as the origin, write position vectors for

B

,

C

and

D

. Prove that the diagonals

AC

and

BD

bisect each other.

Q34Moderate

A circle has centre

O

at the origin and radius

r

. Points

A = -r\mathbf{i}

and

B = r\mathbf{i}

are the ends of a diameter.

P

is any other point on the circle, so

|OP| = r

. Prove that

\angle APB = 90°

.

Q35Challenging

Triangle

ABC

has position vectors

\mathbf{a}

,

\mathbf{b}

,

\mathbf{c}

. The median from

A

goes to the midpoint

M

of

BC

. Let

G

divide

AM

in the ratio

2:1

from

A

. Show that

G = \dfrac{\mathbf{a}+\mathbf{b}+\mathbf{c}}{3}

, and hence that the three medians are concurrent.

Q36Challenging

Let

\mathbf{a}

and

\mathbf{b}

be vectors with

|\mathbf{a}| = |\mathbf{b}|

. Prove that

(\mathbf{a}+\mathbf{b})

and

(\mathbf{a}-\mathbf{b})

are perpendicular.

Q37Challenging

Using vectors in 3D, prove that if

\mathbf{u}

is perpendicular to both

\mathbf{v}

and

\mathbf{w}

, then

\mathbf{u}

is perpendicular to every vector of the form

\alpha\mathbf{v}+\beta\mathbf{w}

for any scalars

\alpha, \beta

.

Worked solutions and answers at openmath.au/year-12/extension-2/further-work-with-vectors