1. 14 Vectors in Three-dimensional Space Case Study
14.1 Vectors in Three-dimensional Rectangular Coordinate System
14.2 Vector Product and Scalar Triple Product
Chapter Summary

2. Case Study
Captain, please follow the following flight route.
A plane is approaching Hong Kong International Airport. The flight-
control operator of the control tower
is trying to give instructions to the
pilot to provide a safe route for landing.
We are now ready for landing. Please indicate a flight route.
To describe the position and the route of
the plane, we can introduce the three-dimensional rectangular coordinate system as shown in the figure.
Let the airport be the origin of the coordinate system. Then we use the triplet (x, y, z) to describe the horizontal position (x, y as in the two-dimensional case) and the height of the plane (z).
The plane is located at the point A(2, 1, 4),
B(−1, −2, 3), C(2, 0, 2), D(1, 1, 1) are points in space, such that the plane follows the route A ® B ® C ® D ® O.

3. 14.1 Vector in Three-dimensional Rectangular Coordinate System
A. Vectors in Three-dimensional Space
For any two points A and B in space, the directed line segment from A to B is called the vector from A to B, and is denoted by The magnitude of is denoted by , which is the same as the vectors on the plane defined before.
The addition, subtraction, scalar multiplication, negative, parallelism of vectors, and the rules of operations of vectors are also defined in the same way as in the case of plane vectors.
For example, for the cube ABCDEFGH, we have
1. (equal vectors)
2. (negative vectors)
3. (parallel vectors)
4. (addition of vectors )
5. (subtraction of vectors)

4. Example 14.1T 14.1 Vector in Three-dimensional
Rectangular Coordinate System
A. Vectors in Three-dimensional Space
Example 14.1T
The figure shows a cube ABCDEFGH. Let = a, = b and = c. Express the following in terms of a, b and c.
(a) (b) (c)
Solution:
(a)
(b)
(c)

5. Example 14.2T 14.1 Vector in Three-dimensional
Rectangular Coordinate System
A. Vectors in Three-dimensional Space
Example 14.2T
The figure shows a cube ABCDEFGH. Prove that
Solution:

6. 14.1 Vector in Three-dimensional Rectangular Coordinate System
B. Representation of Vectors in Three-dimensional
Rectangular Coordinate System
The three-dimensional Cartesian coordinate system R3 consists of three mutually perpendicular axes: x, y and z.
The directions of these axes are aligned in such a way that they obey the right-hand rule.
If the x- and y-axes are represented by the index finger and the middle finger respectively, then the thumb represents the z-axis.

7. 14.1 Vector in Three-dimensional Rectangular Coordinate System
B. Representation of Vectors in Three-dimensional
Rectangular Coordinate System
Every point in space can be represented in the three-dimensional coordinate system by the triplet (x, y, z), where x, y and z represent the directed distances from the yz-, zx- and xy-planes respectively.
These three values are called x-, y- and z-coordinates of the point respectively.
The point of intersection of the three axes is called the origin O and its coordinates are (0, 0, 0).
In the figure, i, j and k are the unit vectors in the positive directions of x-, y- and z-axes respectively. They have a common starting point at the origin, and their terminal points are (1, 0, 0), (0, 1, 0) and (0, 0, 1) respectively.
P(x, y, z) is a point in R3, so we can express the position vector as
By Pythagoras’ theorem, we have

8. 14.1 Vector in Three-dimensional Rectangular Coordinate System
B. Representation of Vectors in Three-dimensional
Rectangular Coordinate System
Now if two points A(x1, y1, z1) and B(x2, y2, z2) in R3 are given, the vector from A to B can be found by subtracting the position vector from which is the same as we did in the case of R2, then we have

9. Example 14.3T 14.1 Vector in Three-dimensional
Rectangular Coordinate System
B. Representation of Vectors in Three-dimensional
Rectangular Coordinate System
Example 14.3T
Given two points P(–3, –2, 8) and Q(0, –5, 4). Find the unit vector in the direction of
Solution:
\ Unit vector

10. 14.1 Vector in Three-dimensional Rectangular Coordinate System
B. Representation of Vectors in Three-dimensional
Rectangular Coordinate System
Consider two vectors pi + qj + rk and si + tj + uk in R3. As i, j and k are non parallel vectors, we have
Property 14.1
pi + qj + rk = si + tj + uk if and only if p = s, q = t and
r = u, and
(b) pi + qj + rk = 0 if and only if p = q = r = 0.

11. Example 14.4T 14.1 Vector in Three-dimensional
Rectangular Coordinate System
B. Representation of Vectors in Three-dimensional
Rectangular Coordinate System
Example 14.4T
Given three points A(–3, 1, 5), B(2, 5, –1) and C(–6, 4, 3). Find the coordinates of a point D if
(a) ABCD forms a parallelogram,
(b) ABDC forms a parallelogram.
Solution:
Let the coordinates of D be (x, y, z).
(a) If ABCD forms a parallelogram,
\ We have
\ The coordinates of D are (11, 0, 9).

12. Example 14.4T 14.1 Vector in Three-dimensional
Rectangular Coordinate System
B. Representation of Vectors in Three-dimensional
Rectangular Coordinate System
Example 14.4T
Given three points A(–3, 1, 5), B(2, 5, –1) and C(–6, 4, 3). Find the coordinates of a point D if
(a) ABCD forms a parallelogram.
(b) ABDC forms a parallelogram.
Solution:
(b) If ABDC forms a parallelogram,
\ We have
\ The coordinates of D are (1, 8, 3).

13. Example 14.5T 14.1 Vector in Three-dimensional
Rectangular Coordinate System
B. Representation of Vectors in Three-dimensional
Rectangular Coordinate System
Example 14.5T
Consider the three vectors a = 3i – 4j + 2k, b = i – 3j – k and c = 5i + 2j + k. If m = –16i – 8j – 7k, express m in terms of a, b and c.
Solution:
Let m = a a + b b + g c.
Consider the determinant of the coefficient matrix:
By Cramer’s rule,

14. Example 14.6T 14.1 Vector in Three-dimensional
Rectangular Coordinate System
B. Representation of Vectors in Three-dimensional
Rectangular Coordinate System
Example 14.6T
Given two points A and B with = 3i – 8j + 5k and
= 6i + 2j – 7k. C is a point on the line segment AB.
Find if
(a) C is the mid-point of AB,
(b) C divides AB in the ratio 2 : 1.
Solution:
(a)
(b)

15. 14.1 Vector in Three-dimensional Rectangular Coordinate System
C. Scalar Product
In the three-dimensional rectangular coordinate system R3, as the unit base vectors i, j and k are mutually perpendicular, we have
i  i = j  j = k  k = 1
i  j = j  i = 0
j  k = k  j = 0
i  k = k  i = 0
We also have the following properties of scalar product:
a  b = b  a
a  (b + c) = a  b + a  c
(ka)  b = k(a  b) = a  (kb)
If a = x1i + y1j + z1k and b = x2i + y2j + z2k are two non-zero vectors, then a  b = x1x2 + y1y2 + z1z2,
where  is the angle between a and b.

16. Example 14.7T 14.1 Vector in Three-dimensional
Rectangular Coordinate System
C. Scalar Product
Example 14.7T
Two vectors r = 2i + 3j – k and s = i + 2k are given.
(a) Find the value of r  s.
(b) Hence find the angle between r and s.
Solution:
(a)
(b)
\ The angle between r and s is 90.

17. Example 14.8T 14.1 Vector in Three-dimensional
Rectangular Coordinate System
C. Scalar Product
Example 14.8T
If the vectors ci + 5j – 3k and 2ci + cj + k are perpendicular to each other, find the value(s) of c.
Solution:
∵ ci + 5j – 3k and 2ci + cj + k are perpendicular to each other.

18. Example 14.9T 14.1 Vector in Three-dimensional
Rectangular Coordinate System
C. Scalar Product
Example 14.9T
Given three points A(2, 1, 6), B(– 5, 3, 5) and C(0, –6, 5) are vertices of DABC. Solve DABC. (Give the answers in surd form or correct to the nearest degree.)
Solution:
(cor. to the nearest degree)

19. Example 14.9T 14.1 Vector in Three-dimensional
Rectangular Coordinate System
C. Scalar Product
Example 14.9T
Given three points A(2, 1, 6), B(– 5, 3, 5) and C(0, –6, 5) are vertices of DABC. Solve DABC. (Give the answers in surd form or correct to the nearest degree.)
Solution:
(cor. to the nearest degree)
(cor. to the nearest degree)

20. Example 14.10T 14.1 Vector in Three-dimensional
Rectangular Coordinate System
C. Scalar Product
Example 14.10T
Given two vectors x = 6j – 5k and y = 3i – 4j.
(a) Find the angle between x and y, correct to the nearest degree.
(b) Find the length of the projection of y on x.
Solution:
(a)
Let  be the angle between x and y.
(cor. to the nearest degree)
\ The angle between x and y is 128.
(b) The length of the projection of y on x

21. 14.2 Vector Product and Scalar Triple Product
A. Definition of Vector Product
Suppose we have two non-zero vectors a and b in the three-dimensional space. The vector product of a and b, denoted by a  b, is the vector which is perpendicular to both a and b, with the magnitude equal to
|a  b| = |a||b|sin,
where  is the angle between a and b (with 0° ≤  ≤ 180°).
Particularly, the direction of a  b is defined in such a way that a, b and a  b always obey the right-hand rule. In conclusion,
a  b = |a||b|sin  ,
where  is the angle between a and b, and is a unit vector whose direction is defined by the right-hand rule.
Note:
1. a  b is read as ‘a cross b’. Therefore the vector product is also
called the cross product.
2. The vector product is only defined in the three-dimensional space.
3. In contrast to the scalar product of two vectors, the vector product is a vector while the scalar product is a scalar.

22. 14.2 Vector Product and Scalar Triple Product
A. Definition of Vector Product
For and two non-zero vectors a and b, a  b = 0 if and only if a and b are parallel to each other.
In particular, if b = a, we have
a  a = 0.
For the unit vectors i, j and k:
i  i = j  j = k  k = 0
i  j = k j  i = k
j  k = i k  j = i
k  i = j i  k = j

23. 14.2 Vector Product and Scalar Triple Product
B. Properties of Vector Product
Property 14.2 Properties of Vector Product
(a) b  a = (a  b)
(b) (a + b)  c = a  c + b  c
(c) a  (b + c) = a  b + a  c
(d) (ka)  b = a  (kb) = k(a  b)
(e) |a  b|2 = |a|2|b|2 – (a  b)2
Proof of (a):
\ b  a = (a  b)

24. 14.2 Vector Product and Scalar Triple Product
B. Properties of Vector Product
Proof of (d):
If k = 0 or a = 0 or b = 0, then
(ka) × b = a × (kb)
= (k a × b) = 0.
Assume that k ¹ 0 and a and b are non-zero.
Let q be the angle between a and b, and be the unit vector in the direction of a × b.
When k > 0,
When k < 0,
Similarly, it can be proved that a × (kb) = k(a × b).
\ (ka) × b = a × (kb) = k(a × b)

25. 14.2 Vector Product and Scalar Triple Product
B. Properties of Vector Product
Proof of (e):
Since |a × b| = |a||b|sin q,
|a × b|2 = (|a||b|sin q)2
= |a|2|b|2sin2q
= |a|2|b|2 − |a|2|b|2cos2q
sin2q = 1 – cos2q
= |a|2|b|2 – (a  b)2
a – b = |a||b|cos q

26. 14.2 Vector Product and Scalar Triple Product
C. Calculation of Vector Product
We can use the determinant to represent the vector product:
If a = x1i + y1j + z1k and b = x2i + y2j + z2k,
then

27. Example 14.11T 14.2 Vector Product and Scalar Triple Product Solution:
C. Calculation of Vector Product
Example 14.11T
For the following pairs of vectors m and n, find the vector products m × n.
(a) m = 3i + 8j, n = 6k
(b) m = –4i + 2j + 6k,
Solution:
(a)
(b)

28. Example 14.12T 14.2 Vector Product and Scalar Triple Product Solution:
C. Calculation of Vector Product
Example 14.12T
P, Q and R are three points with position vectors i + j + k, –2j and –i + 3j – k respectively. Find the unit vectors which are perpendicular to and
Solution:
Unit vectors which are perpendicular to and

29. 14.2 Vector Product and Scalar Triple Product
D. Applications of Vector Product
Consider a parallelogram ABCD.
Area of the parallelogram ABCD =
Since the area of ABD is half that of parallelogram ABCD, we can obtain a formula for the area of triangle:
Area of ABD =
The above formula can be further rewritten as
Area of ABD =

30. Example 14.13T 14.2 Vector Product and Scalar Triple Product Solution:
D. Applications of Vector Product
Example 14.13T
Find the area of the triangle formed by vertices X(2, 1, 1), Y(0, –1, 0) and Z(–2, 1, –1).
Solution:
Area of XYZ

31. 14.2 Vector Product and Scalar Triple Product
E. Scalar Triple Product
The volume of a parallelepiped (a prism with all faces are parallelograms) with sides a, b and c is given by |a  (b × c)|.
The expression a  (b  c) is called the scalar triple product of a, b and c.
Since the scalar product of two vectors is a scalar, thus a  (b  c) is a scalar as the name suggests.
In the three-dimensional rectangular coordinate system, suppose a = x1i + y1j + z1k, b = x2i + y2j + z2k and c = x3i + y3j + z3k, then
Note:
If a  (b  c) = 0, the volume of the parallelepiped with sides a, b and c equals zero. This only when a, b and c are coplanar.

32. 14.2 Vector Product and Scalar Triple Product
E. Scalar Triple Product
Since a determinant is unchanged when interchanging the rows twice and for any non-zero vectors x and y, x  y = y  x, we have the following properties of the scalar triple product:
Property 14.3 Properties of Scalar Triple Product
(a) (a  b)  c = a  (b  c)
(b) a  (b  c) = b  (c  a) = c  (a  b)

33. Example 14.14T 14.2 Vector Product and Scalar Triple Product Solution:
E. Scalar Triple Product
Example 14.14T
If p = 2i + j + 3k, q = 3i – j – 2k and r = –i + 2j – k, find
(a) r × p, and
(b) q  (r × p).
Solution:
(a)
(b)

34. Example 14.15T 14.2 Vector Product and Scalar Triple Product Solution:
E. Scalar Triple Product
Example 14.15T
Consider A(2, 1, 0), B(–3, 4, 5), C(0, –2, 4) and D(1, 2, 5). Find the volume of the parallelepiped with sides , and
Solution:
\ Volume of the parallelepiped

35. Chapter Summary 14.1 Vectors in Three-dimensional
Rectangular Coordinate System
1. Every point in the space can be represented in the
three-dimensional coordinate system by the triplet
(x, y, z), where x, y and z represent the directed distances from the yz-, zx- and xy-planes respectively.
2. The distance between two points A(x1, y1, z1) and B(x2, y2, z2) is given by

36. Chapter Summary 14.1 Vectors in Three-dimensional
Rectangular Coordinate System
1. The rules of operations and properties of vectors in the space are the same as vectors on a plane.
2. In R3, we define three mutually perpendicular unit vectors i, j and k, which point in the positive direction of x-, y- and z-axes respectively.
For a point P(x, y, z) in R3, the position vector can be expressed
as , where

37. Chapter Summary 14.1 Vectors in Three-dimensional
Rectangular Coordinate System
Scalar Product
If a = x1i + y1j + z1k and b = x2i + y2j + z2k, are two non-zero vectors, then
where q is the angle between a and b.

38. Chapter Summary 14.2 Vector Product and Scalar Triple Product
1. If a = x1i + y1j + z1k and b = x2i + y2j + z2k, are non-zero vectors and q is the angle between them, then
2. Area of DABC

39. Chapter Summary 14.2 Vector Product and Scalar Triple Product
1. If a = x1i + y1j + z1k and b = x2i + y2j + z2k and
c = x3i + y3j + z3k are non-zero vectors, then
2. Volume of the parallelepiped with sides a, b and c = |a  (b  c)|.

40. Follow-up 14.1 14.1 Vector in Three-dimensional
Rectangular Coordinate System
A. Vectors in Three-dimensional Space
Follow-up 14.1
The figure shows a cube ABCDEFGH. Let = a, = b and = c. Express the following in terms of a, b and c.
(a) (b) (c)
Solution:
(a)
(b)
(c)

41. Follow-up 14.2 14.1 Vector in Three-dimensional
Rectangular Coordinate System
A. Vectors in Three-dimensional Space
Follow-up 14.2
The figure shows a tetrahedron ABCD. M and N are the mid-points of BC and AD respectively. Prove that
Solution:

42. Follow-up 14.3 14.1 Vector in Three-dimensional
Rectangular Coordinate System
B. Representation of Vectors in Three-dimensional
Rectangular Coordinate System
Follow-up 14.3
Given two points M(–1, 4, 2) and N(3, 1, –3). Find the unit vector in the direction of
Solution:
\ Unit vector

43. Follow-up 14.4 14.1 Vector in Three-dimensional
Rectangular Coordinate System
B. Representation of Vectors in Three-dimensional
Rectangular Coordinate System
Follow-up 14.4
Given three points A(1, 4, −2), B(6, 3, 5) and C(−3, 6, 3). Find the coordinates of a point D if ABCD forms a parallelogram.
Solution:
Let the coordinates of D be (x, y, z).
If ABCD forms a parallelogram,
\ We have
The coordinates of D are (8, 7, 4).

44. Follow-up 14.5 14.1 Vector in Three-dimensional
Rectangular Coordinate System
B. Representation of Vectors in Three-dimensional
Rectangular Coordinate System
Follow-up 14.5
Given three vectors a = – 2i + 4j + k, b = i – 2j + 5k and c = 4i + j – 3k. If r = 14i + 8j, express r in terms of a, b and c.
Solution:
Let r = a a + b b + g c.
Consider the determinant of the coefficient matrix:
By Cramer’s rule,

45. Follow-up 14.6 14.1 Vector in Three-dimensional
Rectangular Coordinate System
B. Representation of Vectors in Three-dimensional
Rectangular Coordinate System
Follow-up 14.6
Given two points A and B with = –5i + 2j – 4k and = 3i + 8j + 8k. C is a point on the line segment AB. Find if
(a) C is the mid-point of AB,
(b) C divides AB in the ratio 3 : 2.
Solution:
(a)
(b)

46. Follow-up 14.7 14.1 Vector in Three-dimensional
Rectangular Coordinate System
C. Scalar Product
Follow-up 14.7
Given two vectors m = i – 2j + 4k and n = 3i + 4j + 3k.
(a) Find the value of m  n.
(b) Hence find the angle between m and n, correct to the nearest degree.
Solution:
(a)
(b) Let  be the angle between m and n.
(cor. to the nearest degree)
\ The angle between m and n is 75.

47. Follow-up 14.8 14.1 Vector in Three-dimensional
Rectangular Coordinate System
C. Scalar Product
Follow-up 14.8
If the vectors bi – 3j + b2k and i + 2bj – 3k are perpendicular to each other, find the value(s) of b.
Solution:
∵ and are perpendicular to each other.

48. Follow-up 14.9 14.1 Vector in Three-dimensional
Rectangular Coordinate System
C. Scalar Product
Follow-up 14.9
Three points A(−3, 3, 4), B(0, −5, 2) and C(4, 1, 7) are vertices of DABC. Solve DABC. (Give the answers in surd form or correct to the nearest degree.)
Solution:

49. Follow-up 14.9 14.1 Vector in Three-dimensional
Rectangular Coordinate System
C. Scalar Product
Follow-up 14.9
Three points A(−3, 3, 4), B(0, −5, 2) and C(4, 1, 7) are vertices of DABC. Solve DABC. (Give the answers in surd form or correct to the nearest degree.)
Solution:
(cor. to the nearest degree)
(cor. to the nearest degree)
(cor. to the nearest degree)

50. Follow-up 14.10 14.1 Vector in Three-dimensional
Rectangular Coordinate System
C. Scalar Product
Follow-up 14.10
Given two vectors a = i + 3j + 5k and b = 2i + 4j + 6k.
(a) Find the angle between a and b, correct to the nearest degree.
(b) Find the length of the projection of a on b.
Solution:
(a)
Let  be the angle between a and b.
(cor. to the nearest degree)
(b) The length of the projection of a on b

51. Follow-up 14.11 14.2 Vector Product and Scalar Triple Product
C. Calculation of Vector Product
Follow-up 14.11
For the following pairs of vectors p and q, find the vector products p × q.
(a) p = 3i + j + 2k, q = i + 4j – k
(b) p = i – 4j – 6k, q = 2i + 5k
(c) p = 2i – 5j + k, q = –6i + 15j – 3k
Solution:
(a)
(b)
(c)

52. Follow-up 14.12 14.2 Vector Product and Scalar Triple Product
C. Calculation of Vector Product
Follow-up 14.12
X, Y and Z are three points with position vectors i – k, 3i + j – 2k and – j + k respectively. Find the unit vectors which are perpendicular to and
Solution:
Unit vectors which are perpendicular to and

53. Follow-up 14.13 14.2 Vector Product and Scalar Triple Product
D. Applications of Vector Product
Follow-up 14.13
Find the area of the triangle with vertices P(0, 1, –1), Q(–2, –1, 1) and R(1, –2, 0).
Solution:
Area of PQR

54. Follow-up 14.14 14.2 Vector Product and Scalar Triple Product
E. Scalar Triple Product
Follow-up 14.14
If x = 3i + j + k, y = –i – j + 3k and z = 2i + 2j – 2k, find
(a) x × y, and
(b) (x × y)  z.
Solution:
(a)
(b)

55. Follow-up 14.15 14.2 Vector Product and Scalar Triple Product
E. Scalar Triple Product
Follow-up 14.15
The coordinates of P, Q, R and S are (0, 0, 2), (3, 0, 4), (1, 2, 3) and (0, 1, –2) respectively. Find the volume of the parallelepiped with sides , and
Solution:
\ Volume of the parallelepiped