Chapter I Vector Quantities
CHAPTER I
VECTOR QUANTITIES
Quantities are anything which can be measured, and stated with number. Quantities in
physics are divided into two types; scalar and vector quantities. Scalar quantities have only
magnitude and completely specified by value and unit. Examples for scalar quantities
such as time, mass, speed, distance, work, power, energy, temperature, etc. For example
time, time is specified by number and its unit, like 1 second, 1 hour, or 1 year. The
operation of a scalar quantity is the same with algebraic operations such as addition,
subtraction and multiplication.
The second type of quantities is vector quantities, quantities which have magnitude and
direction. Vector quantities are expressed by number, unit, and direction. Examples of
vector quantities are displacement, velocity, acceleration, force, momentum, impulse,
electric field, etc. For example, a car has velocity 70 km per hour eastward. The
operations of vector quantities are different with scalar quantities. Vector addition and
multiplication have their own rules, vector rules.
A. Representation of Vector
A vector quantity is symbolized with bold letters or regular letters marked with an arrow
on it and the representation of a vector quantity is represented with an arrow. The
direction of a vector is shown by direction of the arrow and its length of line represents
the magnitude of the vector. For example, look at vector acceleration A and vector force F
below.
F
A
𝛼
Figure 1.1 Vector A and Vector F
Vector A has magnitude written as A or |𝐀| and its direction to the right while vector F
has magnitude written as F or |𝐅| with direction 𝛼 upward of horizontal.
In cartesian system of coordinates, vectors are shown as Figure 2.2. The vectors may be
resolved into its component vectors. Vector S in cartesian coordinates two dimensions
Chapter I Vector Quantities
may be resolved into two component vectors, 𝐒𝐱 and 𝐒𝐲 whereas in three dimensions,
vector T may be resolved into three, 𝐓𝐱 , 𝐓𝐲 and 𝐓𝐳 .
𝑧
𝑦
𝐓𝐳
S
𝐒𝐲
𝑥
𝐒𝐱
(a). Vector S in 2 dimensions
T
𝐓𝐲
𝐓𝐱
𝑦
𝑥
(b). Vector T in 3 dimensions
Figure 1.2 Vectors in cartesian coordinates.
Vector S is formed by two component vectors and written as
𝐒 = 𝐒𝐱 + 𝐒𝐲
Where
𝐒𝐱 = component vector of S in x axis
𝐒𝐲 = component vector of S in y axis
The magnitude of vector S is written as S and the magnitude of its component vectors are
𝑆𝑥 and 𝑆𝑦 . By using pythagorean theorem, it’s found that
𝑆 = √𝑆𝑥 2 + 𝑆𝑦 2
Whereas vector T is formed by three component vectors and written as
𝐓 = 𝐓𝐱 + 𝐓𝐲 + 𝐓𝐳
where
𝐓𝐱 = component vector of T in x axis
𝐓𝐲 = component vector of T in y axis
𝐓𝐳 = component vector of T in z axis
The magnitude of vector T, T may be stated in the form of its component vectors
magnitude 𝑇𝑥 , 𝑇𝑦 , and 𝑇𝑧 . T is derived using Pythagorean Theorem
𝑇 = √𝑇𝑥 2 + 𝑇𝑦 2 + 𝑇𝑧 2
Chapter I Vector Quantities
B. Unit Vector
A unit vector is a dimensionless vector having a magnitude of exactly one. The symbol for
a unit vector is the same with common vector with a hat on it. In cartesian coordinates, it
̂ is a unit vector in positive x
is defined unit vectors corresponding to each axis. 𝐢̂, 𝐣̂, and 𝐤
axis, positive y axis, and positive z axis, respectively, as shown in Figure 1.3.
z
𝑘̂
𝑗̂
𝑖̂
y
𝑥
Figure 1.3 Unit vectors in cartesian coordinates
Thus, by using unit vectors in cartesian coordinates, the unit–vector notation for the
vector S and vector T are
𝐒 = Sx 𝐢̂ + Sy 𝐣̂
̂
𝐓 = 𝑇𝑥 𝐢̂ + 𝑇𝑦 𝐣̂ + 𝑇𝑧 𝐤
Generally, a unit vector of a particular vector is obtained by dividing the vector itself with
̂ and obtained by
its own magnitude. For example, a unit vector of T is symbolized with 𝐓
dividing vector T with T
̂=
𝐓
̂=
𝐓
𝐓
𝑇
̂
𝑇𝑥 𝐢̂ + 𝑇𝑦 𝐣̂ + 𝑇𝑧 𝐤
√𝑇𝑥 2 + 𝑇𝑦 2 + 𝑇𝑧 2
C. Vector Addition
Two vectors or more of the same kind can be added to form a resultant of vector. There
are several methods to add vectors: parallelogram method, triangle method, and
component method.
Chapter I Vector Quantities
1. Parallelogram Method
𝐀
𝐑
𝐀
𝜃
𝜃
𝐁
𝐁
(a)
(b)
Figure 1.4 Addition of vector A and B with parallelogram method
Vector A and B in Figure 1.4 (a) are the same vector so that can be added. To do
parallelogram method, a parallelogram is made by lines A and B, and the diagonal line
between vector A and B is the resultant vector R of A and B, (Figure b).
𝐑=𝐀+𝐁
Since the angle between A and B is 𝜃, you can proof that the magnitude of R using
cosines rule is
𝑅 = √𝐴𝟐 + 𝐵 2 + 2𝐴𝐵 cos 𝜃
2. Triangle and Polynomial Method
Two vectors 𝐀 and 𝐁 that is the same kind in Figure 1.5 (a) will be added. To add vector
𝐁 to vector 𝐀 using triangle method, first we draw vector 𝐀 and then draw vector 𝐁 with
its tail starting from the tip of 𝐀. The resultant vector R, where 𝐑 = 𝐀 + 𝐁 is the vector
drawn from the tail of 𝐀 to the tip of 𝐁, as shown in Figure 1.5 (b).
𝐁
𝐀
𝐑
𝐀
𝐁
𝐑
(a)
(b)
𝐀
𝐁
(c)
Figure 1.5 Triangle method
The order in which 𝐀 and 𝐁 are added is not significant, so that 𝐑 = 𝐀 + 𝐁 = 𝐁 + 𝐀
(Figure c). Furthermore, to add more than two vectors, for example 𝐀 + 𝐁 + 𝐂, polygon
method is used to have resultant vector, 𝐑 = 𝐀 + 𝐁 + 𝐂, as shown in Figure 1.6.
Chapter I Vector Quantities
𝐀
𝐂
𝐁
𝐁
𝐀
𝐂
𝐑
Figure 1.6 Polygon method
When three or more vectors are added, their sum is independent of the way in which the
individual vectors are grouped together. This is called the associative law of addition
𝐀 + (𝐁 + 𝐂) = (𝐀 + 𝐁) + 𝐂
3. Component Method
Now let us see how to use components to add vectors when the graphical method
(parallelogram, triangle, and polygon methods) is not sufficiently accurate. Suppose we
wish to add vector 𝐁 to vector 𝐀. Both of them are vectors having two components in xy
plane. Thus, the unit–vector notation for the vector 𝐀 and 𝐁 are
𝐀 = 𝐴𝑥 𝐢̂ + 𝐴𝑦 𝐣̂
𝐁 = 𝐵𝑥 𝐢̂ + 𝐵𝑦 𝐣̂
𝐑= 𝐀+𝐁
𝐑 = (𝐴𝑥 𝐢̂ + 𝐴𝑦 𝐣̂) + (𝐵𝑥 𝐢̂ + 𝐵𝑦 𝐣̂)
𝐑 = (𝐴𝑥 𝐢̂ + 𝐵𝑥 𝐢̂) + (𝐴𝑦 𝐣̂ + 𝐵𝑦 𝐣̂)
𝐑 = (𝐴𝑥 + 𝐵𝑥 ) 𝐢̂ + (𝐴𝑦 + 𝐵𝑦 ) 𝐣̂
Since 𝐑 in the unit–vector notation is = 𝑅𝑥 𝐢̂ + 𝑅𝑦 𝐣̂, we can see that the components of
the resultant vector are
𝑅𝑥 = 𝐴𝑥 + 𝐵𝑥
𝑅𝑦 = 𝐴𝑦 + 𝐵𝑦
D. Vector Multiplication
If vector 𝐀 is multiplied by a positive scalar quantity 𝑚, then the product 𝑚𝐀 is a vector
that has the same direction as 𝐀 and magnitude 𝑚𝐴. If vector 𝐀 is multiplied by a
negative scalar quantity −𝑚, then the product −𝑚𝐀 has opposite direction to 𝐀 and
magnitude 𝑚𝐴. However, multiplying a vector to another vector is more complicated
than multiplying a vector to a scalar. There are three kinds of vector multiplication, dot
Chapter I Vector Quantities
product, cross product, and dyadic or tensor product. Each product has their own rules.
In this case, we only study about the first two.
1. Dot Product
Dot product of two vectors yields a scalar quantity. Dot product of vector 𝐀 and vector 𝐁
is defined as
𝐀 ∙ 𝐁 = 𝐴𝐵 𝑐𝑜𝑠 𝜃
where 𝜃 is an angle between the two vectors. It tells us that dot product of two
perpendicular vectors is zero. Geometrical interpretation of dot product of vector 𝐀 and
vector 𝐁 implies a scalar multiplication between magnitude of vector 𝐀 with the
projection magnitude of vector 𝐁 on vector 𝐀 (Figure 1.7 a), or a scalar multiplication
between the projection magnitude of vector 𝐀 on vector B with magnitude of vector 𝐁
(Figure 1.7 b).
𝐀
𝐀
𝐵𝐴 = 𝐵 cos 𝜃
𝜃
𝐁
𝜃
𝐁
𝐴𝐵 = 𝐴 cos 𝜃
Figure 1.7 Geometrical interpretation of dot product
If vector 𝐀 and vector 𝐁 are expressed in unit–vector notation, then the dot product is
𝐀 ∙ 𝐁 = (𝐴𝑥 𝐢̂ + 𝐴𝑦 𝐣̂) ∙ (𝐵𝑥 𝐢̂ + 𝐵𝑦 𝐣̂)
𝐀 ∙ 𝐁 = 𝐴𝑥 𝐵𝑥 (𝐢̂ ∙ 𝐢̂) + 𝐴𝑥 𝐵𝑦 (𝐢̂ ∙ 𝐣̂) + 𝐴𝑦 𝐵𝑥 (𝐣̂ ∙ 𝐢̂) + 𝐴𝑦 𝐵𝑦 (𝐣̂ ∙ 𝐣̂)
𝐀 ∙ 𝐁 = 𝐴𝑥 𝐵𝑥 + 𝐴𝑦 𝐵𝑦
Dot product can be used to find the angle between two vectors by using the equation
cos 𝜃 =
𝐀∙𝐁
𝐴𝐵
Some properties of the dot product :
1. 𝐀 ∙ 𝐁 = 𝐁 ∙ 𝐀
̂ ∙𝐤
̂ =1
2. 𝐢̂ ∙ 𝐢̂ = 𝐣̂ ∙ 𝐣̂ = 𝐤
( since 𝜃 = 0 )
̂ = 𝐤
̂ ∙ 𝐢̂ = 0
3. 𝐢̂ ∙ 𝐣̂ = 𝐣̂ ∙ 𝐤
( since 𝜃 = 90𝑜 )
4. 𝐀 and 𝐁 perpendicular if 𝐀 ∙ 𝐁 = 𝟎 and 𝐀 and 𝐁 are not zero
Chapter I Vector Quantities
2. Cross Product
Cross product of two vector quantities yields a vector quantity, and defined as
̂
𝐀 × 𝐁 = 𝐴𝐵 sin 𝜃 𝐧
and its magnitude
|𝐀 × 𝐁| = 𝐴𝐵 sin 𝜃
̂ is a unit vector perpendicular to the
where 𝜃 is an angle between the two vectors and 𝐧
̂ depends on the directions of
plane formed by vector 𝐀 and vector 𝐁. The direction of 𝐧
̂, that is called right hand
vector 𝐀 and vector 𝐁. There is a rule how to find direction of 𝐧
rule. For example, cross product of 𝐢̂ and 𝐣̂, as shown in Figure 1.8.
z
̂
𝐤
𝐣̂
𝐢̂
y
𝑥
Figure 1.8 Cross product of 𝐢̂ and 𝐣̂
̂
𝐢̂ × 𝐣̂ = 1 1 sin 90𝑜 𝐧
̂
𝐢̂ × 𝐣̂ = 𝐧
If we rotate a screw from the tip of vector 𝐢̂ to the tip of vector 𝐣̂, it will move upper in the
̂ . Since the magnitude of 𝐧
̂
̂ = 1 then 𝐧
̂=𝐤
same direction with vector 𝐤
̂
𝐢̂ × 𝐣̂ = 𝐤
Some properties of cross product :
̂ , 𝐣̂ × 𝐤
̂ = 𝐢̂, 𝐤
̂ × 𝐢̂ = 𝐣̂
1. 𝐢̂ × 𝐣̂ = 𝐤
̂, 𝐤
̂ × 𝐣̂ = −𝐢̂, 𝐢̂ × 𝐤
̂ = −𝐣̂
2. 𝐣̂ × 𝐢̂ = −𝐤
̂ ×𝐤
̂ = 0 ( since 𝜃 = 0 )
3. 𝐢̂ × 𝐢̂ = 𝐣̂ × 𝐣̂ = 𝐤
4. 𝐀 × 𝐁 = −𝐁 × 𝐀
5. Vector 𝐀 parallel to vector 𝐁 if 𝐀 × 𝐁 = 0, and 𝐀 and 𝐁 are not zero
Example
̂ and vector 𝐐 = 2𝐢̂ − 2𝐣̂ + 𝐤
̂,
Given vector 𝐏 = 𝐢̂ + 2𝐣̂ + 2𝐤
a. draw vector 𝐏 and 𝐐 in cartesian coordinates
b. find the magnitude of vector 𝐏 and 𝐐
c. find dot product of vector 𝐏 and 𝐐
d. find the product of 𝐏 × 𝐐
e. find the angle between vector 𝐏 and 𝐐
f. find a unit vector perpendicular to the plane formed by vector 𝐏 and 𝐐
Chapter I Vector Quantities
Solution
a. Vector 𝐏 and 𝐐 in cartesian coordinates
z
2
1
2
𝐐
𝐏
2
1
2
𝑥
b. Magnitude of vector 𝐏
𝑃 = √𝑃𝑥 2 + 𝑃𝑦 2 + 𝑃𝑧 2
𝑃 = √12 + 22 + 22
𝑃 = √9
𝑃=3
b. Magnitude of vector 𝐐
𝑄 = √𝑄𝑥 2 + 𝑄𝑦 2 + 𝑄𝑧 2
𝑃 = √22 + (−2)2 + 12
𝑃 = √9
𝑃=3
c. Dot product of vector 𝐏 and 𝐐
̂ ) ∙ (2𝐢̂ − 2𝐣̂ + 𝐤
̂)
𝐏 ∙ 𝐐 = (𝐢̂ + 2𝐣̂ + 2𝐤
𝐏∙𝐐=2−4+4
𝐏∙𝐐=2
d. The product of 𝐏 × 𝐐
̂ ) × (2𝐢̂ − 2𝐣̂ + 𝐤
̂)
𝐏 × 𝐐 = (𝐢̂ + 2𝐣̂ + 2𝐤
̂ + 2𝐣̂ × 2𝐢̂ + 2𝐣̂ × (−2𝐣̂) + 2𝐣̂ × 𝐤
̂
𝐏 × 𝐐 = 𝐢̂ × 2𝐢̂ + 𝐢̂ × (−2𝐣̂) + 𝐢̂ × 𝐤
̂ × 2𝐢̂ + 2𝐤
̂ × (−2𝐣̂) + 2𝐤
̂ ×𝐤
̂
𝐏 × 𝐐 = +2𝐤
̂ − 𝐣̂ − 4𝐤
̂ + 𝟎 + 𝟐𝐢̂ + 𝟒𝐣̂ + 𝟒𝐢̂ + 𝟎
𝐏 × 𝐐 = 𝟎 − 2𝐤
̂
𝐏 × 𝐐 = 𝟔𝐢̂ + 𝟑𝐣̂ − 6𝐤
e. The angle between the two vectors is
|𝐏 × 𝐐|
sin 𝜃 =
𝑃𝑄
sin 𝜃 =
√62 + 32 + (−6)2
3×3
sin 𝜃 =
√81
9
sin 𝜃 = 1
then, 𝜃 = 90𝑜
̂, then
f. Suppose the unit vector is 𝐧
𝐏×𝐐
̂=
𝐧
|𝐏 × 𝐐|
y
Chapter I Vector Quantities
̂
6𝐢̂ + 3𝐣̂ − 6𝐤
𝟗
2
1
2
̂
̂ = 𝐢̂ + 𝐣̂ − 𝐤
𝐧
3
3
3
̂=
𝐧