C h a p t e r 9 Ve c t o r s
443
7 An aeroplane travels on a bearing of 147 degrees for 457 km. Express its position as
a vector in terms of i and j .
˜
˜
8
WORKED
ORKED
9
(3, 4)
~a
0
c
e
b = 4i + 3 j
˜
˜
˜
c = i + 2j –
˜
˜
˜
x
0
~d
x
6k
˜
(3, –4)
d e = – 4i + 3 j
˜
˜
˜
f f = – 2.1i + 3.5 j – 1.4k
˜
˜
˜
10 multiple
ultiple choice
A unit vector in the direction of 6i – 8 j is:
˜
˜
3
4
3
4
----C i– j
A 5i + 5 j
B 5i – 5 j
˜ ˜
˜ ˜
˜ ˜
D
3
------ i
25 ˜
4
-j
– ----25
˜
E none of these
11 Not all unit vectors are smaller than the original vectors. Consider the vector
v = 0.3i + 0.4 j . Show that the unit vector in the direction of v is twice as long as v .
˜
˜
˜
˜
˜
12 Find the unit vector in the direction of w = – 0.1i – 0.02 j .
˜
˜
˜
Math
cad
Example
xample
A ship travels on a bearing of 331 degrees for 125 km. Express its position as a vector
in terms of i and j .
˜
˜
9 Find unit vectors in the direction of the given vector for the following:
a
b
y
y
Unit
vector
in 2D
456
Maths Quest 12 Specialist Mathematics
In summary:
Two vectors are linearly dependent if they are parallel, a = k b , otherwise they are
˜
˜
linearly independent (k ∈ R\{0}).
Any three non-parallel vectors in a plane are also linearly dependent:
a = m b + n c , if m and n are not both zero.
˜
˜
˜
However in three dimensions, three non-parallel vectors are linearly independent
(unless there exist m and n that are not both zero, such that a = mb + nc , which indi˜
˜
˜
cates that a , b and c are linearly dependent).
˜ ˜
˜
WORKED Example 19
Determine the value of p if the following vectors are linearly dependent.
a = i + p j – k, b = i + 3 j + 8k and c = i – j + 2k .
˜
˜
˜
˜
˜ ˜
˜
˜
˜ ˜ ˜
˜
THINK
WRITE
1
Let a = m b + n c .
˜
˜
˜
6
Expand the right-hand side and collect like
components.
Equate the i components.
˜
Express m in terms of n.
Equate the k components.
˜
Substitute m = 1 − n and solve for n.
7
Substitute n = 1.5 into [1].
2
3
4
5
8
9
Equate the j components.
˜
Substitute for m and n to find p.
a = mb + nc
˜
˜
˜
i + p j – k = m ( i + 3 j + 8k ) + n ( i – j + 2k )
˜
˜
˜
˜ ˜
˜
˜ ˜
˜
= ( m + n ) i + ( 3m – n ) j + ( 8m + 2n )k
˜
˜
˜
m+n = 1
m = 1–n
[1]
8m + 2n = – 1
8 ( 1 – n ) + 2n = – 1
8 – 8n + 2n = – 1
– 6n = – 9
n = 1.5
m = 1 – 1.5
m = – 0.5
p = 3m – n
Let
p = 3 ( – 0.5 ) – 1.5
p = –3
Three-dimensional non-zero vectors
If a , b and c are 3-dimensional non-zero vectors:
˜ ˜
˜
1 Show geometrically that: a + b + c ≤ a + b + c
˜ ˜ ˜
˜
˜
˜
2 Prove algebraically that:
a + b ≥ a+b
˜
˜
˜ ˜
a•b ≤ a b
˜ ˜
˜ ˜
Under what circumstances are all of the above statements equal?
3 Prove algebraically that:
458
Maths Quest 12 Specialist Mathematics
5 In the figure at right, a is the vector joining point A to point B, b
˜
˜
joins B to C and c joins C to A.
˜
Prove that a + b + c = 0 .
˜ ˜ ˜
B
~a
A
WORKED
ORKED
Example
xample
17
6 Consider the equilateral triangle at right. Show, using the
properties of vectors only, that line BD, drawn such that BD is
perpendicular to AC, divides AC in half, that is:
~b
~c
C
B
AD = DC .
A
D
C
7 Show that the diagonals of a non-square rhombus intersect at right angles.
(Hint: Make a drawing; a rhombus is a parallelogram with all 4 sides of equal length.)
8 If the length of a vector, a , is given by a , show geometrically that for any two
˜
˜
vectors a and b :
˜
˜
a + b ≥ a+b
˜
˜
˜ ˜
This is known as the triangle inequality.
WORKED
ORKED
Example
xample
18
9 Use vectors to show that the diagonals of a parallelogram bisect each other.
10 Use vectors to show that the angle subtended by a diameter of a circle is a right angle.
11 Use a vector method to show that by joining the midpoints of a parallelogram, the
figure formed is a parallelogram.
12 Consider any two major diagonals of a cube. Use a vector method to show that:
a the diagonals bisect each other
b the acute angle between the diagonals is 70.53°.
WORKED
ORKED
Example
xample
19
13 Determine the value of p if
a = p i + 12 j + 2 k ;
˜
˜
˜
˜
are linearly dependent.
b = i − 3 j + 2k ;
˜ ˜
˜
˜
c = 3i + 2 j + 2k
˜
˜
˜
˜
14 Determine the value of p if
a = 11 i − 3 j + p k ;
˜
˜
˜
˜
are linearly dependent.
b = 2i − 3 j + k ;
˜
˜
˜
˜
c = −i − 3 j − 2k
˜
˜
˜
˜
15 Determine whether the vectors a , b and c are linearly dependent.
˜ ˜
˜
a a = i – 2 j + k ; b = 2i + j – 3k ; c = – i + 2 j – 2k
˜
˜
˜
˜
˜
˜
˜
˜
˜
˜
˜
˜
b a = – 5 j + k ; b = 3i + 4 j – 5k ; c = 2i + j – 3k
˜
˜
˜ ˜
˜ ˜
˜
˜ ˜ ˜
˜
C h a p t e r 9 Ve c t o r s
469
10 Find the equation of the paths described by each of the following vector functions:
4t
t2 + 4
a u = -----------------2- i + -----------------2- j (Hint: Add x -and y-components and factorise.)
˜
(t + 2) ˜ (t + 2) ˜
What is the initial position of u ?
˜
When does x = y? What is the position at this time?
26
ET
SHE
Work
Example
xample
Graphics Calculator tip! Vector functions of time
1. Make sure the MODE settings are for radians and parametric plotting.
O
Suppose the path of an object is given by r = 2 cos t i + 3 sin t j , 0 < t < 2.
˜
˜
˜
Vector
2. Press Y= and at X1T= enter 2 cos(T), and at Y1T= enter 3 sin(T).
functions of
time
3. Press WINDOW and set Tmin= to 0, Tmax= to 2π, Tstep= to 0.1 and then press GRAPH .
(You may need to alter the other window settings to get a good view – this window uses
ZDecimal.)
4. The path appears to be that of an ellipse – some algebra will verify this. Press TRACE
to verify that the object starts at (2, 0). (You can change the plotting speed by changing
Tstep.)
CASI
WORKED
ORKED
b u = ( t + 2 )i + ( t + 1 ) 2 j
˜
˜
˜
c u = ( 2 cos t + 3 )i + ( 3 sin t – 1 ) j
˜
˜
˜
11 Ship A’s position vector is given by u = ( 3t + 1 )i + ( 4t – 2 ) j . Ship B has a position
˜ ships’ paths
˜ cross.
vector of v = ( 2t + 3 )i + ( 5t + 1 ) j . ˜Find where the
˜
˜
˜
12 Let u = t i + t 2 j . Find the equation of the path. Consider vector v = e t i + e 2t j .
˜
˜
Show˜ that ˜the path
of this vector is the same as u ’s path. Assuming˜ both ˜vectors’
˜
equations start when t = 0, do these vectors’ ever coincide?
5. Note that the plot may not look correct in a poor window. For example, the left screen
below shows the plot in the window [–5, 5] by [–5, 5]; it looks more like a circle than
an ellipse. To make sure that you have a true proportion scale in a plot like this where
it is especially important, use ZOOM and 5: ZSquare (see right screen below).
9.2
C h a p t e r 9 Ve c t o r s
473
Using the vectors a = i – 2 j and b = 2i + 3 j answer questions 9 and 10.
˜
˜
˜
˜
˜
˜
9 The scalar resolute of a on b is given by:
˜4 ˜
4
4
B – --------D – -----A – ----C – 4--E −4
13
5
13
5
10 The vector resolute of b parallel to a is:
˜
˜
4
A – -----B – 4--5- ( i – 2 j )
5
˜
˜
4
4
-( i – 2 j)
--- ( 2i + 3 j )
D – ----E
–
13 ˜
5 ˜
˜
˜
C
4
– ------ ( 2i
13 ˜
+ 3 j)
˜
11 Let u ( t ) be a position vector of an object, whose position varies with time. If
˜
u = 3 sin t i + 2 cos t j then the path this object takes is:
˜
˜
˜
A a straight line
B a parabola
C a circle
D an ellipse
E unable to be determined with the given information
9E
9E
9F
Short answer
1 A fire observation tower reaches 40 m above the ground. Susan is 400 m from the tower,
which is at a bearing of 60° (N 60° E) from her. State the position vector from Susan’s
current position to the top of the tower.
9A,B
40 m
N
400 m
60°
Susan
E
2 A boat sails 5 km due east from H, turns northward at a bearing of 45° (N 45° E) for a
distance of 10 km and then travels due north for a further 5 km to point X.
a Find the position vector from H to X.
b Find the distance from H to X (correct to 2 decimal places).
9B
3 Let u = 4i + 3 j and v = –i + 2 j . Find the angle between the two vectors, in radians, to
˜
˜
˜ ˜
˜
˜
4 decimal places.
9C
4 Let u = 3i – 5 j and v = –4i + j . Find:
˜
˜
˜
˜
˜
a u+v
b u–v ˜
c u•v
d û
˜
˜ ˜
˜ ˜
˜ ˜
e the angle between u and v .
˜
˜
5 Find the angles that the vector v = 3i – 2 j + 4k makes with the x, y and z axes.
˜
˜
˜
˜
6 Find value(s) of p, such that pi + 2 ( 1 – 3 p ) j is perpendicular to 2 pi + 3 j .
˜
˜
˜
˜
9A–C
9C
9C
474
9D
9D
9D
9E
9F
Maths Quest 12 Specialist Mathematics
7 Show that the diagonals of a square intersect at right angles.
8 Show that for any two vectors u and v ,
˜( u + v )˜• ( u – v ) = u 2 – v 2
˜ ˜
˜ ˜
˜
˜
9 Determine the value of p if a , b and c are linearly dependent where:
˜
˜
˜
a = 6i – 4 j + 3k ; b = – 4i + p j – 2k; c = 5i + 2 j – 4k
˜
˜
˜
˜
˜
˜ ˜
˜
˜
˜
˜
˜
10 Let u = 2i + 3 j – k and v = i + j – 2k .
˜
˜
˜ ˜
˜
˜
˜ ˜
a Find a unit vector parallel to u .
˜
b Resolve v into components parallel
and perpendicular to u .
˜
˜
1
11 Find the equation of the path of the time-varying position vector u = --- i + 2 ( t 2 – 1 ) j . State
t˜
˜
˜
the type of path (linear, parabolic, etc.). Hence, sketch its graph.
Analysis
1 A radar station tracks a jet fighter flying with constant speed. If the radar station is considered to be at the origin, the fighter’s starting position is 2i + 8 j + k and 1 minute later it
˜
˜ ˜
is at 8i – 4 j + 13k . The units are in kilometres.
˜
˜
a State the˜ vector which indicates the path of the fighter.
b State a unit vector in the direction of this path.
c Find a vector, in terms of a parameter m, which represents the position of the fighter at
any time along the path.
d Find the point along the path where the fighter is closest to the station.
e Find the distance from the station at this point.
f Find the speed of the plane in km/h.
g How fast is the plane rising (or falling)?
C h a p t e r 9 Ve c t o r s
475
2 Consider the box shown below.
The coordinates (in cm) of point D are (3, 0, 4), while the coordinates of E are (0, 5.5, 4).
z
E
D
y
a
b
c
d
e
f
g
x
C
Find the coordinates of point C.
Express the line joining C to E as a vector.
Show that the two diagonals in the same plane as CE intersect with an angle of 73.7°.
Find the volume of the box in litres.
Express the longest diagonal, from the origin, as a vector.
Find the length of this diagonal.
Find the angle that this diagonal makes with the other long diagonal from D.
3 The parallelogram OXYZ has O at the origin. The vector joining O to Z is given by 5i
˜
while the vector joining O to X is given by 2i + 7 j .
˜
˜
a Sketch the parallelogram, labelling all vertices.
b State the vectors joining Z to Y and Y to X.
c State the vectors which represent the diagonals of the parallelogram.
d Find the cosine of the angle between the diagonals. Express your answer in simplest
surd form.
e Find the angle that OX makes with the x-axis.
f State the vector resolute of the vector joining O to X in the direction of OZ.
g Let P be a point on the extended line of XY, such that the vector joining P to Z is
perpendicular to OY. Find the coordinates of P.
h Find the area of the parallelogram.
4 A river flows west-east at 5 m/s. A
swimmer, in still water, can swim 3 m/s
and tries to swim directly across the
river from south to north.
a Draw a vector diagram to illustrate
this situation.
b Find the resultant speed of the
swimmer.
c Find the bearing of the swimmer.
d If it took the swimmer 2 minutes to
reach the opposite bank, how wide is
the river?
e How far downstream would the
swimmer be carried?
f Repeat parts b to e if the swimmer
had started on the north bank.