STPM Mathematics T / A Level
Vectors
M.K.Lim
September 6, 2012
M.K.Lim
STPM Mathematics T / A Level
Representation of Vectors
B
A
A vector is a quantity which has magnitude and
specific direction in space. A quantity with magnitude but no
direction is called a scalar.
~ to show the displacement from A to B.
We write as AB
~ can be called vector a
Displacement is move from A to B as AB
Note the arrowhead points(direction) from A towards B.
M.K.Lim
STPM Mathematics T / A Level
Equal Vectors
• Two vectors with same magnitude and same direction are
equal. a = b.
• It follows that they can be represented by any line of right
length and direction.
• In this case, both vectors has same direction and length.
b
a
M.K.Lim
STPM Mathematics T / A Level
Negative vectors
• If two vectors a and b ,have the same magnitude but opposite
directions, we say a = −b
~ = −DC
~
• An other useful notation is valid such as CD
b
a
M.K.Lim
STPM Mathematics T / A Level
Subtraction of Vectors
• If we wish to subtract two vectors a and b, we can expressed
as
• a − b =a + (-b)
We say the subraction of vectors can be considered as the addition
of a reversed vector b.
It is easier to add a reversed vector form of b.
M.K.Lim
STPM Mathematics T / A Level
Modulus of a Vector
• The modulus of a vector is its magnitude.
• It is written as |a|. This is the length of the line represented.
• Given vector a = 3î + 4ĵ + 5k̂ ,
• Then modulus |a| is
√
32 + 42 + 52
M.K.Lim
STPM Mathematics T / A Level
Scalar Multiplication of a Vector
• If λ is positive real number , then λ is a vector in the same
direction as a and of magnitude λa.
• It is natural that −λa is in opposite direction.
M.K.Lim
STPM Mathematics T / A Level
Example of Scalar Multiplication
D
B
A
C
~ is twice as long as vector AB.
~
• Vector CD
~ = 2AB
~
• Represent it by CD
• We say λ is 2.
• Scalar means magnitude is involved, direction is not.
M.K.Lim
STPM Mathematics T / A Level
The Addition of Vectors - Triangle Law
This law is important for solving problems.
q
B
C
p
p+q
Consider 4 ABC. A
• Vector for p for side AB and Vector for q for side BC
• Resultant Vector is p + q represented by side AC
• Note that the arrow point towards C for resultant vector
p + q.
M.K.Lim
STPM Mathematics T / A Level
Addition Law Triangle Law Contd...
• Its the head-to-tail story...
~ + BC
~ = AC
~
• AB
• If side AB represents vector p
• Side BC represented by vector q
• Then side AC is the resultant, as p + q going from tail of p to
head of q.
• Note :The tail of vector q follows the head of vector p
M.K.Lim
STPM Mathematics T / A Level
Addition Law Using Parallelogram ABCD
B
b
C
a
a
A
D
b
• Parallel sides AB and DC represented by vector a
• Similarly, parallel sides BC and AD represented by vector b
~ =a+b
• In the triangle ABC, resultant AC
~ = a+ b
• In triangle ADC , AC
• Therefore a + b = b + a
• Since AC is the common between 4 ABC and 4 ACD
M.K.Lim
STPM Mathematics T / A Level
Diagonals of a Parallelogram
R
Q
b
a+b
O
a−b
a
P
~
~
~ = a+b
Given OP = a, OR = b, then OQ
~
~
~ = a + (−b) = a − b from
Looking at 4 PQR: RP = RQ + QP
subtraction of Triangle Law.
~ = - RP
~ = - (a- b) = b - a .
Also PR
These are important to be remembered:
• (a − b) is the vector from endpoint of b to to the
endpoint of a.
• (b − a) is the vector from endpoint of a to the endpoint
of b.
M.K.Lim
STPM Mathematics T / A Level
Diagonals in a Parallelogram
Consider 4 OPR, we have
• Solid black line is vector (a − b)
• Dashed black line is vector (b − a)
b
a−b
a
M.K.Lim
STPM Mathematics T / A Level
Vectors Illustrated in Cartesian coordinates
j
5
4
A
3
2
C
1
0
1
2
3
i
4
~ = 3î + 4ĵ and vector b is OC
~ = î + 2ĵ
• Vector a is OA
• Resultant vector arrow :Aligning head of vector a with the tail
of vector b.
• So it becomes a + b = a + b
~ + OC
~ = CA
~
• OA
M.K.Lim
STPM Mathematics T / A Level
Area of a Parallelogram Using Vectors
j
6
5
4
D
C
~
3 AD
2
h
1
A
θ
1
2B 3
i
4
~
AB
M.K.Lim
STPM Mathematics T / A Level
Determinant Method to compute Cross product
• Use determinant method to solve 2D matrix
• Area by determinant method should yield the same answer of
8 units squared
2 0
2 4
M.K.Lim
STPM Mathematics T / A Level
Angle between two vectors
• Angle between two vectors is unique labelled as θ.
• Two vectors a and b are shown with angle in between.
• It is the angle between the directions when the both lines
converge or diverge from a point shown as a blue dot. It is
only angle θ and not any other.
a
θ
b
M.K.Lim
STPM Mathematics T / A Level
Unit Vector
• Given a is a vector
• The unit vector is written as â
• A unit vector is a vector whose length is 1, so magnitude of a
is 1
a
|a|
• A unit vector is in the direction of v is vector over its
magnitude
• Definition: â =
• Applied to Cartesian coordinates, i is the unit vector in Ox
direction and j is the unit vector in Oy direction
M.K.Lim
STPM Mathematics T / A Level
Scalar Product (Dot)
• The scalar product of two vectors a and b is defined as
ab cos θ
• where θ is the angle between them
• a.b = ab cos θ
• Sometimes Scalar Product is also known as Dot Product
M.K.Lim
STPM Mathematics T / A Level
Vector Product (Cross)
• The vector product of two vectors a and b is defined as
ab sin θ where θ is the angle between them
• |a × b| = ab sin θ
• Sometimes the Vector Product is also known as the Cross
Product
• This product acts in a direction perpendicular to both a and b
M.K.Lim
STPM Mathematics T / A Level
Vector Product of two vectors a and b
• Two vectors are parallel, θ = 0◦ , then |a × b| = 0
• Two vectors are perpendicular θ = 90◦ , then |a × b| = ab
M.K.Lim
STPM Mathematics T / A Level
Parallel Vectors
Properties of Scalar Product
b
π
a
• Two vectors a and b are parallel, then ab = ab cos π
• Then a.b = - a.b since cos 180◦ = -1
• For like parallel vectors a.b = ab
• For unlike parallel vectors a.b = - ab
• when a = b, then a.b = a.a = a2
• In Cartesian unit vectors i,j,k we have i.i = j.j = k.k = 1
M.K.Lim
STPM Mathematics T / A Level
Perpendicular Vectors
• When two vectors a and b are perpendicular, then the dot
product of them is a.b = 0
• Because cos 90◦ = 0
• For unit vectors, we have i.j = j.k = k.i = 0
b
a
M.K.Lim
STPM Mathematics T / A Level
Cartesian Unit Vectors
• Now if î is the unit vector in direction of Ox
• Now if ĵ is the unit vector in direction of Oy
• Now if k̂ is the unit vector in direction of Oz
y
x
z
M.K.Lim
STPM Mathematics T / A Level
Vector Equation of a Line
• The equation of a line can be expressed in two forms
• Vector form
• Cartesian form
M.K.Lim
STPM Mathematics T / A Level
Vector Equation of a Line
• We want to find the vector equation of the blue line shown as
AP.
• This line is parallel to a direction vector b which shows the
direction
• Recall the straight line equation y = mx + c
• Similarly, we can use vectors to find the equation of a line
• Consider a line parallel to vector b which passes through a
fixed point A with position vector a
• Vector b is the direction vector for the line
• We shall see the development of r = a + λb
M.K.Lim
STPM Mathematics T / A Level
Vector Equation of a Line Contd...
~ then AP
~ = λb
• If r is the position vector OP
• where λ is a scalar parameter. Relationships of length is
shown by λ
~ = OA
~ + AP
~
• Now OP
~
• Therefore we have r = a + λb since r is OP
• This equation gives the position of one point on the line
• That is P is on the line ⇔ r = a + λb
r = a + λb
M.K.Lim
STPM Mathematics T / A Level
Vector Equation of a Line
y
A
a
b
x
O
r
P(x, y , z)
r = a + λb
M.K.Lim
STPM Mathematics T / A Level
Example of Vector equation of a Line
A line whose vector equation is r = (5î − 2ĵ + 4k̂) +
λ(2î − ĵ + 3k̂) is parallel to vector 2î − ĵ + 3k̂ and is passes
through the point whose position vector is 5î − 2ĵ + 4k̂.
r is the position vector of any point P.
x î + y ĵ + z k̂ = 5î − 2ĵ + 4k̂ + λ(2î − ĵ + 3k̂)
= (5 + 2λ)î + (−2 − λ)ĵ + (4 + 3λ)k̂


x
Equating coefficients from above, we have ∴ λ = y


z
M.K.Lim
STPM Mathematics T / A Level
= 5 + 2λ
= −2 − λ
= 4 + 3λ
Example of Vector equation of a Line Contd ...
x −5
2
y +2
∴λ=
−1
z −4
∴λ=
3
∴λ=
So, the Cartesian form of a vector equation of a line is
x −5
y +2
z −4
=
=
2
−1
3
M.K.Lim
STPM Mathematics T / A Level
Cartesian Equation of a Line
y
a
A
x î + b ĵ + c k̂
x
O
r
P(x, y , z)
z
M.K.Lim
STPM Mathematics T / A Level
General Vector Equation of a Line
If a line passes through A(x1 , y1 , z1 ) and is parallel to aî + bĵ + ck̂
its equation may be written as
r = (x1 î + y1 ĵ + z1 k̂) + λ(aî + b ĵ + c k̂)
x = x1 + λa
y = y1 + λb
z = z1 + λc
M.K.Lim
STPM Mathematics T / A Level
V.Equation Cartesian form
In Cartesian format, it is shown as
λ=
x − x1
y − y1
z − z1
=
=
a
b
c


x = x1 + λa
λ = y = y1 + λb


z = z1 + λc
Note that the point A (x1 , y1 , z1 ) is one of the infinite set of points
on the line. Hence the equations representing a given line is not
unique.
M.K.Lim
STPM Mathematics T / A Level
Equations of a Plane
• Two types namely, Vector and Cartesian form
M.K.Lim
STPM Mathematics T / A Level
Vector Equation of a Plane
Plane ( green ) is defined as distance d from origin O and is
perpendicular to unit vector n̂ shown.
N
x
P
r
d
n̂
O
M.K.Lim
STPM Mathematics T / A Level
Standard form of Vector equation of a Plane
If line ON is perpendicular to the plane then, for any point P on
the plane , NP is perpendicular to ON.
~ = d n̂.Since P is on the plane,it
If r is position vector of P,then ON
~ ON
~ =0
means that NP.
The equation is called the scalar product form of the vector
equation of a plane.
~ = r − d n̂.
If r is a position vector,then NP
Therefore it becomes (r - dn̂).dn̂ = 0
This implies that r.n̂ − dn̂.n̂ = 0
But n̂.n̂ = 1,
So that means
r.n̂ = d
The equation is the standard form of a vector of a plane.
M.K.Lim
STPM Mathematics T / A Level
Cartesian form of Plane
Using r.n̂ = d idea and n̂ = lx + my + nz
Now, if P is a point (x, y , z) on this plane, its position vector
r = xî + yĵ + zk̂ , satisfies the equation r.n̂ = d.
So that (x î + y ĵ + z k̂).(lx + my + nz) = d
lx + my + nz = d
Example:Find the Cartesian equation of this plane r.(2i+3j-4k) = 1
Solution:
Comparing with r.n̂ = d
(2i + 3j − 4k) means n̂= lx + my + nz = 1.
here d = 1. so l = 2, m = 3, n = −4 therefore
2x + 3y − 4z = 1
M.K.Lim
STPM Mathematics T / A Level