REVIEW SET 14A
NON-CALCULATOR
1 Using a scale of 1 cm represents 10 units, sketch a vector to represent:
a an aeroplane taking off at an angle of 8± to a runway with a speed of 60 m s¡1
b a displacement of 45 m in a north-easterly direction.
¡! ¡!
¡! ¡! ¡!
2 Simplify:
a AB ¡ CB
b AB + BC ¡ DC.
3 Construct vector
equations for:
a
l
b
j
q
¡
!
4 If PQ =
5
µ
m
k
r
p
n
¶
µ
¶
µ
¶
¡!
¡
!
¡
!
¡4
¡1
2
, RQ =
, and RS =
, find SP.
1
2
¡3
p
O
[BC] is parallel to [OA] and is twice its length.
Find, in terms of p and q, vector expressions for:
¡!
¡¡!
a AC
b OM.
A
M
q
C
B
0
1
0
1
3
¡12
6 Find m and n if @ m A and @ ¡20 A are parallel vectors.
n
2
0
1
0
1
2
¡6
¡!
¡!
¡!
7 If AB = @ ¡7 A and AC = @ 1 A, find CB.
4
¡3
µ
¶
µ
¶
µ
¶
3
¡3
¡1
8 If p =
, q=
, and r =
, find:
a p²q
¡2
4
5
b q ² (p ¡ r)
9 Consider points X(¡2, 5), Y(3, 4), W(¡3, ¡1), and Z(4, 10). Use vectors to show that
WYZX is a parallelogram.
b is a right angle.
10 Consider points A(2, 3), B(¡1, 4), and C(3, k). Find k if BAC
11 Explain why:
b the expression a ² b £ c does not need brackets.
µ
¶
¡4
12 Find all vectors which are perpendicular to the vector
.
5
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a a ² b ² c is meaningless
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13 In this question you may not assume any diagonal
properties of parallelograms.
¡!
OABC is a parallelogram with OA = p and
¡!
OC = q. M is the midpoint of [AC].
A
p
a Find in terms of p and q:
¡!
¡¡!
i OB
ii OM
b Hence show that O, M, and B are collinear, and
that M is the midpoint of [OB].
M
O
14 Find the values of k such that the following are unit vectors:
C
q
4
7
1
k
a
15 Suppose j a j = 2, j b j = 4, and j c j = 5.
µ ¶
k
b
k
a
a a²b
b b²c
c a²c
Find:
B
c
60°
b
16 Find a and b if J(¡4, 1, 3), K(2, ¡2, 0), and L(a, b, 2) are collinear.
0
1
1
17 Given j u j = 3, j v j = 5, and u £ v = @ ¡3 A, find the possible values of u ² v.
¡4
18 [AB] and [CD] are diameters of a circle with centre O.
¡!
¡!
a If OC = q and OB = r, find:
¡!
¡!
i DB in terms of q and r
ii AC in terms of q and r.
b What can be deduced about [DB] and [AC]?
0
1
0
1
2¡t
t
a Find t given that @ 3 A and @ 4 A are perpendicular.
t
t+1
19
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b Show that K(4, 3, ¡1), L(¡3, 4, 2), and M(2, 1, ¡2) are vertices of a right angled
triangle.
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REVIEW SET 14B
CALCULATOR
1 Copy the given vectors and find geometrically:
b y ¡ 2x
a x+y
y
x
2 Show that A(¡2, ¡1, 3), B(4, 0, ¡1), and C(¡2, 1, ¡4) are vertices of an isosceles triangle.
µ ¶
µ
¶
4
¡3
and s =
find:
a jsj
b jr + sj
c j 2s ¡ r j
3 If r =
1
2
µ
¶
µ
¶ µ
¶
¡2
3
13
4 Find scalars r and s such that r
+s
=
.
1
¡4
¡24
5 Given P(2, 3, ¡1) and Q(¡4, 4, 2), find:
¡
!
a PQ
b the distance between P and Q
c the midpoint of [PQ].
6 If A(4, 2, ¡1), B(¡1, 5, 2), C(3, ¡3, c) are vertices of triangle ABC which is right angled
at B, find the value of c.
0
1
0
1
2
¡1
7 Suppose a = @ ¡3 A and b = @ 2 A. Find x given a ¡ 3x = b.
1
3
8 Find the angle between the vectors a = 3i + j ¡ 2k and b = 2i + 5j + k.
9 Find two points on the Z-axis which are 6 units from P(¡4, 2, 5).
µ
¶
µ 2
¶
3
t +t
10 Determine all possible values of t if
and
are perpendicular.
3 ¡ 2t
¡2
11 Prove that P(¡6, 8, 2), Q(4, 6, 8), and R(19, 3, 17) are collinear.
0
1
0
1
¡4
¡1
a u²v
b the angle between u and v.
12 If u = @ 2 A and v = @ 3 A, find:
1
¡2
13 [AP] and [BQ] are altitudes of triangle ABC.
¡!
¡!
¡!
Let OA = p, OB = q, and OC = r.
¡!
¡!
a Find vector expressions for AC and BC in terms of p,
q, and r.
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b Using the property a ² (b ¡ c) = a ² b ¡ a ² c,
deduce that q ² r = p ² q = p ² r.
c Hence prove that [OC] is perpendicular to [AB].
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Q
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O
A
B
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14 Find two vectors of length 4 units which are parallel to 3i ¡ 2j + k.
b
15 Find the measure of DMC.
D(1, -4,¡3)
C(3, -3,¡2)
A(-2,¡1, -3)
B(2,¡5, -1)
M
0
1
0 1
2
1
16 Find all vectors perpendicular to both @ ¡1 A and @ 1 A.
¡2
2
0
17
B
a Find k given that @
1
k
C
A is a unit vector.
p1
2
¡k
0
1
3
b Find the vector which is 5 units long and has the opposite direction to @ 2 A.
¡1
0
1
0
1
2
¡1
18 Find the angle between @ ¡4 A and @ 1 A.
3
3
b M given that M is the
19 Determine the measure of QD
midpoint of [PS].
P
M
S
Q
R
7 cm
A
B
10 cm
C
D
4 cm
REVIEW SET 14C
¡
! ¡! ¡!
¡
! ¡!
1 Find a single vector which is equal to:
a PR + RQ
b PS + SQ + QR
0
1
0
1
0
1
6
2
¡1
2 For m = @ ¡3 A, n = @ 3 A, and p = @ 3 A, find:
1
¡4
6
a m¡n+p
b 2n ¡ 3p
c jm + pj
3 What geometrical facts can be deduced from the equations:
¡!
¡!
¡!
¡!
a AB = 12 CD
b AB = 2AC ?
4 Given P(2, ¡5, 6) and Q(¡1, 7, 9), find:
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a the position vector of Q relative to P
c the distance from P to the X-axis.
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5
¡
!
¡!
In the figure alongside, OP = p, OR = r, and
¡!
RQ = q. M and N are the midpoints of [PQ] and
[QR] respectively.
Find, in terms of p, q, and r:
¡!
¡!
¡!
¡¡!
a OQ
b PQ
c ON
d MN
Q
P
M
p
N q
O
r
R
0
1
0
1
0 1
¡3
2
3
6 Suppose p = @ 1 A, q = @ ¡4 A, and r = @ 2 A. Find x if:
2
1
0
a p ¡ 3x = 0
b 2q ¡ x = r
7 Suppose j v j = 3 and j w j = 2. If v is parallel to w, what values might v ² w take?
8 Find a unit vector which is parallel to i + rj + 2k and perpendicular to 2i + 2j ¡ k.
0
1
0
1
¡4
t
9 Find t if @ t + 2 A and @ 1 + t A are perpendicular vectors.
t
¡3
10 Find all angles of the triangle with vertices K(3, 1, 4), L(¡2, 1, 3), and M(4, 1, 3).
0 1
µ 5 ¶
k
13
b @kA
11 Find k if the following are unit vectors:
a
k
k
b
12 Use vector methods to find the measure of GAC
in the rectangular box alongside.
E
H
F
G
5 cm
D
A
µ
13 Using p =
8 cm
B
C
4 cm
¶
µ
¶
µ
¶
3
¡2
1
, q=
, and r =
, verify that:
¡2
5
¡3
p ² (q ¡ r) = p ² q ¡ p ² r.
14 P(¡1, 2, 3) and Q(4, 0, ¡1) are two points in space. Find:
¡
!
¡
!
a PQ
b the angle that PQ makes with the X-axis.
µ
¶
µ
¶
¡¡!
¡!
¡! ¡
!
¡!
¡
!
¡2
5
, MP =
, MP ² PT = 0, and j MP j = j PT j.
15 Suppose OM =
4
¡1
¡!
Write down the two possible position vectors OT.
16 Given p = 2i ¡ j + 4k and q = ¡i ¡ 4j + 2k, find the angle between p and q.
17 Suppose u = 2i + j, v = 3j, and µ is the acute angle between u and v.
Find the exact value of sin µ.
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18 Find two vectors of length 3 units which are perpendicular to both ¡i + 3k and 2i ¡ j + k.
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REVIEW SET 15A
µ
1 For the line that passes through (¡6, 3) with direction
NON-CALCULATOR
¶
4
, write down the corresponding:
¡3
a vector equation
b parametric equations
c Cartesian equation.
µ ¶ µ
¶
µ
¶
x
18
¡7
=
+t
. Find m.
2 (¡3, m) lies on the line with vector equation
y
¡2
4
µ
¶
µ ¶
3
2
3 Line L has equation r =
+t
.
¡3
5
a Locate the point on the line corresponding to t = 1.
µ
¶
4
b Explain why the direction of the line could also be described by
.
10
c Use your answers to a and b to write an alternative vector equation for line L.
4 P(2, 0, 1), Q(3, 4, ¡2), and R(¡1, 3, 2) are three points in space.
a Find parametric equations of line (PQ).
20
b
b Show that if µ = PQR,
then cos µ = p p
26 33
.
5 Triangle ABC is formed by three lines:
µ ¶ µ
¶
µ ¶
µ ¶ µ ¶
µ
¶
x
4
1
x
7
1
=
+t
.
Line (BC) is
=
+s
.
Line (AB) is
y
¡1
3
y
4
¡1
µ ¶ µ
¶
µ ¶
x
¡1
3
Line (AC) is
=
+u
. s, t, and u are scalars.
y
0
1
a Use vector methods to find the coordinates of A, B, and C.
¡!
¡!
¡!
b Find j AB j, j BC j, and j AC j.
c Classify triangle ABC.
6
a Consider two unit vectors a and b. Prove that the vector a + b bisects the angle between
vector a and vector b.
b Consider the points H(9, 5, ¡5), J(7, 3, ¡4), and K(1, 0, 2).
Find the equation of the line L that passes through J and bisects HbJK.
c Find the coordinates of the point where L meets (HK).
7 Suppose A is (3, 2, ¡1) and B is (¡1, 2, 4).
a Write down a vector equation of the line through A and B.
¡!
b Find the equation of the plane through B with normal AB.
p
c Find two points on (AB) which are 2 41 units from A.
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8 For C(¡3, 2, ¡1) and D(0, 1, ¡4), find the coordinates of the point where the line passing
through C and D meets the plane with equation 2x ¡ y + z = 3.
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p
¡!
¡!
9 Suppose OA = a, OB = b, j a j = 3, j b j = 7, and a £ b = i + 2j ¡ 3k. Find:
a a²b
b the area of triangle OAB.
a How far is X(¡1, 1, 3) from the plane x ¡ 2y ¡ 2z = 8?
10
b Find the coordinates of the foot of the perpendicular from Q(¡1, 2, 3) to the line
2 ¡ x = y ¡ 3 = ¡ 12 z.
a Find if possible the point where the line through L(1, 0, 1) and M(¡1, 2, ¡1) meets
the plane with equation x ¡ 2y ¡ 3z = 14.
11
b Find the shortest distance from L to the plane.
12 The equations of two lines are:
L1 : x = 3t ¡ 4, y = t + 2, z = 2t ¡ 1
L2 : x =
y¡5
¡z ¡ 1
=
.
2
2
a Find the point of intersection of L1 and the plane 2x + y ¡ z = 2.
b Find the point of intersection of L1 and L2 .
c Find the equation of the plane that contains L1 and L2 .
13 Show that the line x ¡ 1 =
y+2
z¡3
=
2
4
is parallel to the plane 6x + 7y ¡ 5z = 8 and
find the distance between them.
p
14 x2 + y 2 + z 2 = 26 is the equation of a sphere with centre (0, 0, 0) and radius 26 units.
Find the point(s) where the line through (3, ¡1, ¡2) and (5, 3, ¡4) meets the sphere.
15 When an archer fires an arrow, he is suddenly aware of a breeze which pushes his shot off-target.
The speed of the shot j v j is not affected by the wind, but the arrow’s flight is 2± off-line.
a Draw a vector diagram to represent the situation.
b Hence explain why:
i the breeze must be 91± to the intended direction of the arrow
ii the speed of the breeze must be 2 j v j sin 1± .
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16 In the figure ABCD is a parallelogram. X is
B(4,¡4,-2)
the midpoint of BC, and Y is on [AX] such
¡!
¡!
that AY = 2YX.
Y
a Find the coordinates of X and D.
b Find the coordinates of Y.
A(1,¡3,-4)
c Show that B, Y, and D are collinear.
8
x¡y+z =5
<
2x + y ¡ z = ¡1
17 Solve the system
:
7x + 2y + kz = ¡k for any real number k
using row operations.
Give geometric interpretations of your results.
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C(10,¡2,¡0)
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REVIEW SET 15B
CALCULATOR
1 Find the vector equation of the line which cuts the y-axis at (0, 8) and has direction 5i + 4j.
p
2 A yacht is sailing with constant speed 5 10 km h¡1 in the direction ¡i ¡ 3j. Initially it is at
point (¡6, 10). A beacon is at (0, 0) at the centre of a tiny atoll. Distances are in kilometres.
a Find in terms of i and j :
i the initial position vector of the yacht
ii the direction vector of the yacht
iii the position vector of the yacht at any time t hours, t > 0.
b Find the time when the yacht is closest to the beacon.
c If there is a reef of radius 8 km around the atoll, will the yacht hit the reef?
3 Write down i a vector equation ii parametric equations for the line passing through:
µ
¶
4
b (¡1, 6, 3) and (5, ¡2, 0).
a (2, ¡3) with direction
¡1
4 A small plane can fly at 350 km h¡1 in still conditions.
Its pilot needs to fly due north, but needs to deal with
a 70 km h¡1 wind from the east.
a In what direction should the pilot face the plane
in order that his resultant velocity is due north?
b What will the speed of the plane be?
5 Find the angle between line L1 passing through (0, 3) and (5, ¡2), and line L2 passing through
(¡2, 4) and (¡6, 7).
µ
¶
1
at exactly 2:17 pm.
6 Submarine X23 is at (2, 4). It fires a torpedo with velocity vector
¡3
µ
¶
¡1
Submarine Y18 is at (11, 3). It fires a torpedo with velocity vector
at 2:19 pm to
a
intercept the torpedo from X23. Distance units are kilometres. t is in minutes.
a Find x1 (t) and y1 (t) for the torpedo fired from submarine X23.
b Find x2 (t) and y2 (t) for the torpedo fired from submarine Y18.
c At what time does the interception occur?
d What was the direction and speed of the interception torpedo?
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7 Suppose P1 is the plane 2x ¡ y ¡ 2z = 9 and P2 is the plane x + y + 2z = 1.
L is the line with parametric equations x = t, y = 2t ¡ 1, z = 3 ¡ t.
b P1 and P2 .
Find the acute angle between:
a L and P1
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x¡8
y+9
z ¡ 10
=
=
and L2 : x = 15 + 3t, y = 29 + 8t,
8 Consider the lines L1 :
3
¡16
7
z = 5 ¡ 5t.
a Show that the lines are skew.
b Find the acute angle between them.
c Line L3 is a translation of L1 which intersects L2 . Find the equation of the plane containing
L2 and L3 .
d Find the shortest distance between them.
a Find the equation of the plane through A(¡1, 2, 3), B(1, 0, ¡1), and C(0, ¡1, 5).
9
b If X is (3, 2, 4), find the angle that (AX) makes with this plane.
a Find all vectors of length 3 units which are normal to the plane x ¡ y + z = 6.
b Find a unit vector parallel to i + rj + 3k and perpendicular to 2i ¡ j + 2k.
10
c The distance from A(¡1, 2, 3) to the plane with equation 2x ¡ y + 2z = k
Find k.
is 3 units.
11 Find the angle between the lines with equations 4x ¡ 5y = 11 and 2x + 3y = 7.
12 Consider A(2, ¡1, 3) and B(0, 1, ¡1).
a Find the vector equation of the line through A and B.
b Hence find the coordinates of C on (AB) which is 2 units from A.
13 Find the angle between the plane 2x + 2y ¡ z = 3 and the line
x = t ¡ 1, y = ¡2t + 4, z = ¡t + 3.
14 Let r = 2i ¡ 2j ¡ k, s = 2i + j + 2k, t = i + 2j ¡ k, be the position vectors of the
points R, S, and T, respectively. Find the area of the triangle RST.
15 Classify the following line pairs as either parallel, intersecting, or skew. In each case find the
measure of the acute angle between them.
a x = 2 + t, y = ¡1 + 2t, z = 3 ¡ t and x = ¡8 + 4s, y = s, z = 7 ¡ 2s
b x = 3 + t, y = 5 ¡ 2t, z = ¡1 + 3t and x = 2 ¡ s, y = 1 + 3s, z = 4 + s
0
1
0
1
1
2
16 p = @ ¡1 A and q = @ 3 A
2
¡1
0
1
0 1
a Find p £ q.
1
2
b Find m if p £ q is perpendicular to the line L with equation r = @ ¡2 A + ¸ @ 1 A.
3
m
c Hence find the equation of the plane P containing L which is perpendicular to p £ q.
d Find t if the point A(4, t, 2) lies on the plane P .
e For the value of t found in d, if B is the point (6, ¡3, 5), find the exact value of the sine
of the angle between (AB) and the plane P .
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a Show that the plane 2x + y + z = 5 contains the line
L1 : x = ¡2t + 2, y = t, z = 3t + 1, t 2 R .
b For what values of k does the plane x + ky + z = 3 contain L1 ?
c Hence find the values of p and q for which the following system of equations has an infinite
number of solutions. Clearly explain your reasoning.
8
2x + y + z = 5
<
x¡y+z =3
:
¡2x + py + 2z = q
5
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<
18 Consider the system
x ¡ 3y + 2z = ¡5
3x + y + (2 ¡ k)z = 10
:
¡2x + 6y + kz = 5 where k can take any real value.
a Reduce the system to echelon form.
b For what value of k does the system have no solutions? Interpret this result geometrically.
c
i For what value(s) of k does the system have a unique solution?
ii Find the unique solution in terms of k, and interpret the result geometrically.
iii Find the unique solution when k = 1.
REVIEW SET 15C
1 Find the velocity vector of an object moving in the direction 3i ¡ j with speed 20 km h¡1 .
2 A moving particle has coordinates P(x(t), y(t)) where x(t) = ¡4 + 8t and y(t) = 3 + 6t.
The distance units are metres, and t > 0 is the time in seconds. Find the:
a initial position of the particle
b position of the particle after 4 seconds
c particle’s velocity vector
d speed of the particle.
3 Trapezium KLMN is formed by the following lines:
µ ¶ µ ¶
µ
¶
µ ¶ µ
¶
µ
¶
x
2
5
x
33
¡11
(KL) is
=
+p
.
(ML) is
=
+q
.
y
19
¡2
y
¡5
16
µ ¶ µ ¶
µ ¶
µ ¶ µ
¶
µ
¶
x
3
4
x
43
¡5
(NK) is
=
+r
.
(MN) is
=
+s
.
y
7
10
y
¡9
2
p, q, r, and s are scalars.
a Which two lines are parallel? Explain your answer.
b Which lines are perpendicular? Explain your answer.
c Use vector methods to find the coordinates of K, L, M, and N.
d Calculate the area of trapezium KLMN.
4 Find the angle between the lines:
L1 : x = 1 ¡ 4t, y = 3t
L2 : x = 2 + 5s, y = 5 ¡ 12s.
and
5 Consider A(3, ¡1, 1) and B(0, 2, ¡2).
¡!
a Find j AB j.
b Show that the line passing through A and B can be described by
r = 2j ¡ 2k + ¸(¡i + j ¡ k) where ¸ is a scalar.
cyan
magenta
95
yellow
Y:\HAESE\IB_HL-3ed\IB_HL-3ed_15\476IB_HL-3ed_15.cdr Thursday, 5 April 2012 5:07:35 PM BEN
100
50
75
25
0
5
95
100
50
75
25
0
5
95
100
50
75
25
0
5
95
100
50
75
25
0
5
c Find the angle between (AB) and the line with vector equation t(i + j + k).
black
IB_HL-3ed
6 Let i represent a displacement 1 km due east
and j represent a displacement 1 km due north.
Road A passes through (¡9, 2) and (15, ¡16).
Road B passes through (6, ¡18) and (21, 18).
y
B
20
(21,¡18)
H(4,¡11)
10
(-9,¡2)
a Find a vector equation for each of the roads.
b An injured hiker is at (4, 11), and needs to travel
the shortest possible distance to a road.
Towards which road should he head, and how far
will he need to walk to reach this road?
-20
-10
10
x
20
-10
(15,-16)
-20
(6,-18)
A
7 Given the points A(4, 2, ¡1), B(2, 1, 5), and C(9, 4, 1):
¡!
¡!
a Show that AB is perpendicular to AC.
b Find the equation of the line through:
i A and B
ii A and C.
8 The triangle with vertices P(¡1, 2, 1), Q(0, 1, 4), and R(a, ¡1, ¡2) has area
Find a.
p
118 units2 .
9 Consider A(¡1, 2, 3), B(2, 0, ¡1), and C(¡3, 2, ¡4).
a Find the equation of the plane defined by A, B, and C.
b
b Find the measure of CAB.
b C is a right angle. Find r.
c D(r, 1, ¡r) is a point such that BD
10 Given A(¡1, 2, 3), B(1, 0, ¡1), and C(1, 3, 0), find:
a the normal vector to the plane containing A, B, and C
b D, the fourth vertex of parallelogram ACBD
c the area of parallelogram ACBD
d the coordinates of the foot of the perpendicular from C to the line AB.
11 P(2, 0, 1), Q(3, 4, ¡2), and R(¡1, 3, 2) are three points in space. Find:
¡
! ¡!
¡!
a PQ, j PQ j, and QR
b the parametric equations of (PQ)
c a vector equation of the plane PQR.
12 Given the point A(¡1, 3, 2), the plane 2x ¡ y + 2z = 8, and the line defined by
x = 7 ¡ 2t, y = ¡6 + t, z = 1 + 5t, find:
a the distance from A to the plane
b the coordinates of the point on the plane nearest to A
c the shortest distance from A to the line.
13
a Find the equation of the plane through A(¡1, 0, 2), B(0, ¡1, 1), and C(1, 2, ¡1).
b Find the equation of the line, in parametric form, which passes through the origin and is
normal to the plane in a.
cyan
magenta
95
yellow
Y:\HAESE\IB_HL-3ed\IB_HL-3ed_15\477IB_HL-3ed_15.cdr Wednesday, 2 May 2012 2:29:36 PM BEN
100
50
75
25
0
5
95
100
50
75
25
0
5
95
100
50
75
25
0
5
95
100
50
75
25
0
5
c Find the point where the line in b intersects the plane in a.
black
IB_HL-3ed
14 Consider the lines with equations
x¡3
z+1
=y¡4=
2
¡2
and
x = ¡1 + 3t, y = 2 + 2t, z = 3 ¡ t.
a Are the lines parallel, intersecting, or skew? Justify your answer.
b Determine the acute angle between the lines.
15 Line 1 has equation
x¡8
y+9
z ¡ 10
=
=
.
3
¡16
7
0 1 0 1
0
1
x
15
3
Line 2 has vector equation @ y A = @ 29 A + ¸ @ 8 A.
z
5
¡5
a Show that lines 1 and 2 are skew.
b Line 3 is a translation of line 1 which intersects line 2. Find the equation of the plane
containing lines 2 and 3.
c Hence find the shortest distance between lines 1 and 2.
d Find the coordinates of the points where the common perpendicular meets the lines 1 and 2.
16 Lines L1 and L2 are defined by
0
1
0
1
0
1
0
1
3
¡1
3
¡1
L1 : r = @ ¡2 A + s @ 1 A and L2 : r = @ 0 A + t @ ¡1 A.
¡2
2
¡1
1
a Find the coordinates of A, the point of intersection of the lines.
b Show that the point B(0, ¡3, 2) lies on the line L2 .
c Find the equation of the line BC given that C(3, ¡2, ¡2) lies on L1 .
d Find the equation of the plane containing A, B, and C.
e Find the area of triangle ABC.
f Show that the point D(9, ¡4, 2) lies on the normal to the plane passing through C.
17 Three planes have the equations given below:
Plane A: x + 3y + 2z = 5
Plane B: 2x + y + 9z = 20
Plane C:
x ¡ y + 6z = 8
cyan
magenta
95
yellow
Y:\HAESE\IB_HL-3ed\IB_HL-3ed_15\478IB_HL-3ed_15.cdr Friday, 20 April 2012 9:07:52 AM BEN
100
50
75
25
0
5
95
100
50
75
25
0
5
95
100
50
75
25
0
5
95
Show that plane A and plane B intersect in a line L1 .
Show that plane B and plane C intersect in a line L2 .
Show that plane A and plane C intersect in a line L3 .
Show that L1 , L2 , and L3 are parallel but not coincident.
What does this mean geometrically?
100
50
75
25
0
5
a
b
c
d
e
black
IB_HL-3ed
913
ANSWERS
REVIEW SET 14A
1 a
REVIEW SET 14B
1
a
b
60 m¡s-1
x
y
y
8°
y - 2x
x+y
Scale: 1 cm º 10 m¡s-1
-x
b
-x
p
p
2 AB = AC = 53 units and BC = 46 units
) 4 is isosceles
p
p
p
3 a
13 units
b
10 units
c
109 units
Scale: 1 cm º 10 m
45 m
N
4 r = 4, s = 7
45°
5
2
¡
!
a AC
¡!
b AD
3
a q=p+r
b l=k¡j+n¡m
a p+q
5
b
8
¡8
7
7
3
p
2
+
1
q
2
6 m = 5, n =
a ¡13
8
b ¡36
0
³ ´
¡6
1
3
b
¡ 12
A
8 64:0±
p
46 units
a k = § p7
b k = § p1
15
a a ² b = ¡4
b b ² c = 10
33
1
p
2
+
18 ¼ 80:3±
1
q
2
2
c a ² c = ¡10
16 a = ¡2, b = 0
p
p
17 If µ is acute, u ² v = 199; if µ is obtuse, u ² v = ¡ 199.
¡!
b LM =
cyan
5
¡3
¡4
¡!
, KM =
¡2
¡2
¡1
magenta
25
0
5
95
100
50
75
25
0
b = 90±
M
5
25
0
5
95
100
50
75
25
0
5
)
b DB = AC, [DB] k [AC]
95
a t = ¡4
50
19
ii r + q
75
a
100
i q+r
18
b ¡ p5
14
16 k
3
2
¡1
0
¡2
1
19 ¼ 26:4±
REVIEW SET 14C
¡
!
¡
!
1 a PQ
b PR
3
¡3
11
b
7
¡3
¡26
2
a
3
a AB =
1
CD,
2
4
¡
!
a PQ =
¡3
12
3
5
a r+q
b ¡p + r + q
95
14
ii
yellow
Y:\HAESE\IB_HL-3ed\IB_HL-3ed_an\913IB_HL-3ed_an.cdr Thursday, 17 May 2012 2:28:23 PM BEN
100
i p+q
a
a k = § 12
17
50
13
15 ¼ 16:1±
14
75
5
, t 6= 0
4
9 (0, 0, 1) and (0, 0, 9)
or ¡3
12 a 8
b ¼ 62:2±
¡!
¡
!
a AC = ¡p + r, BC = ¡q + r
13
³ ´
50
3
6 c=
2
3
10 t =
10 k = 6
a a ² b is a scalar, so a ² b ² c is a scalar dotted with a
vector, which is meaningless.
b b £ c must be done first otherwise we have the cross product
of a scalar with a vector, which is meaningless.
12 t
1
)
2
¡ 23
14 § p4 (3i ¡ 2j + k)
11
c (¡1, 3 12 ,
1
1
5
7 @ ¡3
1
4
4
a
black
[AB] k [CD]
b
c
p
74 units
b C is the midpoint of [AB].
p
162 units
c r+
c
1
q
2
p
61 units
d ¡ 12 p +
1
r
2
IB HL 3ed
914
ANSWERS
0
8
a
1
p
i
5
1
3
2
3
1
2
p
k
5
+
13
1
¡10
b x=
2
p
9 t=2§ 2
A
7 v ² w = §6
p31
110
15
a
target
11
a k=
14
¡
!
a PQ =
³ ´
¡
!
15 OT =
b k=
12 ¼ 40:7±
b ¼ 41:81±
³
or
2
p
5
17 sin µ =
§ p1
3
2
¡2
´
16 ¼ 61:6±
3
7
1
18 § p3
59
a X(7, 3, ¡1), D(7, 1, ¡2)
3
¡!
¡!
¡3
c BD =
and BY =
0
16
x
y
6
5
2
+s
³
4
10
´
REVIEW SET 15B
³ ´
a
x
y
z
a1
a2
x
y
z
a
3
2
¡1
=
17
3
A,
b ( 83 ,
a §7
b
16
p
14
a ( 15 ,
¡1
6
3
=
a
b
c
d
, ¸2R
7
a ¼ 15:8±
8
28:6±
p
14
2
³
´
4
, t2R
¡1
6
¡8
¡3
+t
, t2R
10
b
p1 i
74
+
p8 j
74
+
b ¼
p
¡p 3
p3
¡ 3
p3 k
74
d 14 units
64:1±
or ¡ p1 i ¡
74
p8 j
74
¡
p3 k
74
c k = ¡7 or 11
x
y
z
a
50
25
0
c 2x + 3y + 6z = 147
11 72:35± or 107:65±
5
95
100
50
5 8:13±
b ¼ 65:9±
a 4x + 2y + z = 3
p
p3
a
and
¡p 3
3
9
b ¼ 343 km h¡1
X23, x1 = 2 + t, y1 = 4 ¡ 3t, t > 0
Y18, x2 = 13 ¡ t, y2 = 3 ¡ 2a + at, t > 2
interception occurred at 2:22:30 pm
bearing ¼ 13± west of south, ¼ 4:54 units per minute
b ¼
units2
75
25
0
5
95
+t
6
c 6x ¡ 8y ¡ 5z = ¡35
100
50
75
25
0
5
95
100
50
75
25
´
¸2R
7 4
, )
3 3
magenta
2
¡3
ii x = ¡1 + 6t, y = 6 ¡ 8t, z = 3 ¡ 3t, t 2 R
12
cyan
³
a 11:5± east of due north
b
b (¡1, 3, 1)
=
4
units
17 9
, )
5 5
x
y
z
i
C
a They do not meet, the line is parallel to the plane.
b
x
y
i
ii x = 2 + 4t, y = ¡3 ¡ t, t 2 R
c (¡5, 2, 9) or (11, 2, ¡11)
9
units
1
3
1
3
¡4
0
5
+¸
b ¡4x + 5z = 24
a
³ ´
0
+¸@
a
a
0 1
7
3
¡4
=
8 (6, ¡1, ¡10)
0
5
, ¸2R
4
3
c (7, 3 34 , ¡3 14 )
5
³ ´
+¸
a A(5, 2), B(6, 5), C(8, 3)
p
p
p
¡
!
¡
!
¡
!
b j AB j = 10 units, j BC j = 8 units, j AC j = 10 units
c isosceles
a¡+¡b
a OABC is a rhombus.
A
So, its diagonals
b
B
bisect its angles.
b
12
0
8
i ¡6i + 10j
ii ¡5i ¡ 15j
iii (¡6 ¡ 5t)i + (10 ¡ 15t)j, t > 0
b t = 0:48 h
c shortest distance ¼ 8:85 km, so will miss reef
b
11
³ ´
2
O
10
=
a x = 2 + t, y = 4t, z = 1 ¡ 3t, t 2 R
¡
!
¡
!
j PQ ² QR j
b Use cos µ = ¡
! ¡
!
j PQ jj QR j
a
7
x
y
1
yellow
Y:\HAESE\IB_HL-3ed\IB_HL-3ed_AN\914IB_HL-3ed_an.cdr Tuesday, 15 May 2012 9:15:42 AM BEN
95
5
³ ´
¡
!
¡
!
So, BD = 3BY, etc.
, ¡1).
meet at the point ( 43 , ¡ 14
3
75
4
=
b Y(5, 3, ¡2)
1
¡1
0
17 If k = ¡2, the planes meet in the line
x = 43 , y = ¡ 11
+ t, z = t, t 2 R . If k 6= ¡2 the planes
3
b x = ¡6 + 4t, y = 3 ¡ 3t, t 2 R
c 3x + 4y = ¡6
2 m = 10
³ ´
³ ´
4
2
3 a (5, 2)
b
is a non-zero scalar multiple of
10
5
³ ´
i isosceles triangle ) 2 remaining angles = 89± each,
breeze makes angle of 180 ¡ 89 = 91± to intended
direction of the arrow.
1
speed
ii bisect angle 2± and use sin 1± = 2
jvj
) speed = 2 jvj sin 1±
b
REVIEW SET 15A
³ ´ ³ ´ ³ ´
x
¡6
4
1 a
=
+t
, t2R
y
3
¡3
c
|v|
actual
flight
intended
direction
2°
5
¡2
¡4
4
8
breeze
|v|
b ¼ 123:7± , b
b = 45±
10 K
L ¼ 11:3± , M
§ 12
13
14 (4, 1, ¡3) and (1, ¡5, 0)
units
100
6
x=@
¡1
black
=
2
¡1
3
+¸
¡2
2
¡4
, ¸2R
IB HL 3ed
918
ANSWERS
³
b
2¡
p2 ,
6
¡1 +
2
p
,
6
3¡
p4
6
2+
p2 ,
6
¡1 ¡
2
p
,
6
3+
p4
6
³
13 7:82±
p
9 2
2
14
´
and
´
a intersecting at (4, 3, 1), angle ¼ 44:5±
b skew, angle ¼ 71:2±
16
a 5
¡1
1
1
b m=1
d t=2
17
18
e
c x¡y¡z =0
b k = ¡1
c t(p + 10) = q + 2 has infinitely many solutions for t when
p + 10 = 0 and q + 2 = 0, ) p = ¡10, q = ¡2
1
0
0
a
¡3
10
0
2
¡4 ¡ k
k+4
¡5
25
¡5
b No solutions if k = ¡4. Two planes are parallel and
intersected by the third plane.
c
i Unique solution when k 6= ¡4.
¡5
10
, y = 2, z =
iii (3, 2, ¡1)
ii x = 1 +
k+4
k+4
Planes meet at a point.
REVIEW SET 15C
p
1 2 10(3i ¡ j)
³ ´
2
a (¡4, 3)
b (28, 27)
3
a (KL) is parallel to (MN) as
8
6
c
³
5
¡2
12
13
a 5x + y + 4z = 3
³
is parallel to
b µ ¼ 27:0±
14
a intersecting at (¡1, 2, 3)
15
b 2x + 3y + 6z = 147
c 14 units
d (5, 7, 3) on line 1 and (9, 13, 15) on line 2
16
a A(2, ¡1, 0)
c r=
d 3x ¡ y + 2z = 7
f normal is
17
x
y
z
0
¡3
2
e
=
3
¡2
¡2
+u
p
3
1
¡4
14 units2
+¸
3
¡1
2
e The three planes have no common point of intersection. The
line of intersection of any two planes is parallel to the third
plane.
´ ³
´
¡5
2
´
4
=0
10
³ ´ ³ ´
4
¡5
and (NK) is perpendicular to (MN) as
²
=0
10
2
b (KL) is perpendicular to (NK) as
5
¡2
b x = 5t, y = t, z = 4t, t 2 R
1 2
, )
14 7
d 10 m s¡1
´
³
¡4
¡1
4
b x = 2 + ¸, y = 4¸, z = 1 ¡ 3¸, ¸ 2 R
¡4
1
2
x
4
0 +¸
y
+ ¹ ¡1 , ¸, ¹ 2 R
=
c
4
¡3
1
z
p
a 3 units
b (1, 2, 4)
c
116 units
5
c ( 14
,
p4
114
p
¡
!
¡
!
, j PQ j = 26 units, QR =
¡
!
a PQ =
units2
15
1
4
¡3
11
²
c K(7, 17), L(22, 11), M(33, ¡5), N(3, 7) d 261 units2
4 30:5±
p
¡
!
5 a j AB j = 27 units
b A lies on the line r where ¸ = ¡3 and B lies on r where
¸ = 0 ) the line between A and B is the same as line r,
so it can be described by r.
c 70:5±
³ ´ ³ ´
³ ´
x
¡9
4
6 a Road A:
=
+¸
, ¸2R
y
2
¡3
³ ´ ³
´
³ ´
x
6
5
Road B:
=
+¹
, ¹2R
y
¡18
12
b Road B, 13 km
¡2
¡1
6
¡
! ¡
!
a AB ² AC =
5
2
2
=0
i x = 4 ¡ 2t, y = 2 ¡ t, z = ¡1 + 6t, t 2 R
ii x = 4 + 5s, y = 2 + 2s, z = ¡1 + 2s, s 2 R
8 a = 4 or ¡ 36
5
p
2§ 10
2
c r=
c ¼ 11:8 units2
b D(¡1, ¡1, 2)
cyan
magenta
95
yellow
Y:\HAESE\IB_HL-3ed\IB_HL-3ed_an\918IB_HL-3ed_an.cdr Thursday, 17 May 2012 2:29:55 PM BEN
100
50
75
25
0
95
50
75
25
0
5
95
100
50
75
25
0
5 2
, )
6 3
95
50
75
25
0
d ( 16 ,
5
5
¡1
3
a n=
5
10
b ¼ 55:9±
a 14x + 29y ¡ 4z = 32
100
9
5
b
²
100
7
black
IB HL 3ed