1.3 Vectors and Scalars
Representation of a vector
A vector is represented by a line with an arrow at its end or at its middle, as shown by the two equal vectors a to the right.
The arrow indicates the direction.
The length indicates the magnitude.
Components of a vector
A vector may be decomposed into one vertical and one horizontal component, as follows:
A vector’s magnitude may be found from its vertical and horizontal components, through the formula (known as Pythagoras Theorem): A^2 = Ax^2 = Ay^2 .
A vector’s direction (angle with the horizontal) may be calculated by means of the following formula: θ = tan^-1 Ay/Ax
Vector manipulation
When two or more vectors are added or subtracted, a resultant vector is formed.
Figure 1.3.8 – Component forces of a block on a slope
SCALARS AND VECTORS
Scalars : The physical quantities which have only magnitude but no direction, are called scalar quantities.
For example – distance, speed, work, temperature, mass, etc.
Scalars are added, subtracted, multiplied and divided by ordinary laws of algebra.
Vectors : For any quantity to be a vector,
- it must have magnitude.
- it must have direction.
- it must satisfy parallelogram law of vector addition.
For example – displacement, velocity, force, etc.
Note – Electric current has magnitude as well as direction but still it is not treated as a vector quantity because it is added by ordinary law of algebra.
TYPES OF VECTORS
LIKE VECTORS
Vectors having same direction are called like vectors. The magnitude may or may not be equal.
and 
are like vectors. These are also called parallel vectors or collinear vectors.
EQUAL VECTORS
Vectors having same magnitude and same direction are called equal vectors.
Here and are equal vectors 
and are equal vectors Thus, equal vector is a special case of like vector.
UNLIKE VECTORS
Vectors having exactly opposite directions are called unlike vectors.
The magnitude may or may not be equal.
and 
are unlike vectors.
NEGATIVE VECTORS
Vectors having exactly opposite direction and equal magnitudes are called negative vectors.
Here and are negative vectors, 
and are negative vectors, Thus negative vectors is a special case of unlike vectors.
UNIT VECTOR
Vector which has unit magnitude. It represents direction only. For example take a vector
. Unit vector in the direction of 
is 
, which is denoted as
. 
, is read as “B cap” or “B caret”.
. Unit vector in the direction of is , which is denoted as . , is read as “B cap” or “B caret”.
ORTHOGONAL UNIT VECTOR
A set of unit vectors, having the directions of the positive x, y and z axes of three dimensional rectangular coordinate system are denoted by
. They are called orthogonal unit vectors because angle between any of the two unit vectors is 90º.
. They are called orthogonal unit vectors because angle between any of the two unit vectors is 90º.
The coordinate system which has shown in fig. is called right handed coordinate system. Such a system derives its name from the fact that right threaded screw rotated through 90º from OX to OY will advance in positive Z direction as shown in the figure.
NULL VECTOR (ZERO VECTOR)
A vector of zero magnitude is called a zero or null vector. Its direction is not defined. It is denoted by 0.
Properties of Null or Zero Vector :
- The sum of a finite vector and the zero vector is equal to the finite vector
i.e., 
- The multiplication of a zero vector by a finite number n is equal to the zero vector
i.e., 0 n = 0
- The multiplication of a finite
by a zero is equal to zero vector
i.e., 
AXIAL VECTOR
Vector associated with rotation about an axis i.e., produce rotation effect is called axial vector. Examples are angular velocity, angular momentum, torque etc.
COPLANAR VECTORS
Vectors in the same plane are called coplanar vectors.
POSITION VECTORS AND DISPLACEMENT VECTORS
The vector drawn from the origin of the coordinate axes to the position of a particle is called position vector of the particle. If A (x1, y1, z1) and B (x2, y2, z2) be the positions of the particle at two different times of its motion w.r.t. the origin O, then position vector of A and B are
The displacement vector is
= 
LAWS OF VECTOR ALGEBRA
(Commutative law of addition)
(Associative law of addition)
ADDITION OF VECTORS
TRIANGLE LAW OF VECTOR ADDITION
It states that if two vectors acting on a particle at the same time are represented in magnitude and direction by the two sides of a triangle taken in one order, their resultant vector is represented in magnitude and direction by the third side of the triangle taken in opposite order.
Magnitude of 
is given by 
is given by where 
is the angle between 
and
.
is the angle between and.Direction of 
: Let the resultant 
makes an angle 
with the direction of 
. Then from right angle triangle QNO,
: Let the resultant makes an angle with the direction of . Then from right angle triangle QNO, 
- |
| is maximum, if cosθ = 1, θ = 0° (parallel vector)
Rmax 
= A + B
= A + B- |
| is minimum, if cosθ = –1, θ = 180° (opposite vector)
Rmin 
- If the vectors A and B are orthogonal,
i.e., 
PARALLELOGRAM LAW OF VECTOR ADDITION
It states that if two vectors are represented in magnitude and direction by the two adjacent sides of a parallelogram then their resultant is represented in magnitude and direction by the diagonal of the parallelogram.
Let the two vectors 
and
, inclined at angle 
are represented by sides 
of parallelogram OPQS, then resultant vector 
is represented by diagonal 
of the parallelogram.
and, inclined at angle are represented by sides of parallelogram OPQS, then resultant vector is represented by diagonal of the parallelogram.
If 
< 90° , (acute angle) 
=
+
,
is called main (major) diagonal of parallelogram
< 90° , (acute angle) =+, is called main (major) diagonal of parallelogramIf 
> 90° , (obtuse angle) 
=
+
,
is called minor diagonal.
> 90° , (obtuse angle) =+, is called minor diagonal.POLYGON LAW OF VECTOR ADDITION
If a number of non zero vectors are represented by the (n–1) sides of an n-sided polygon then the resultant is given by the closing side or the nth side of the polygon taken in opposite order.
So, 
or, 
Note:
- Resultant of two unequal vectors cannot be zero.
- Resultant of three coplanar vectors may or may not be zero.
- Minimum no. of coplanar vectors for zero resultant is 2 (for equal magnitude) and 3 (for unequal magnitude).
- Resultant of three non coplanar vectors cannot be zero. Minimum number of non coplanar vectors whose sum can be zero is four.
- Polygon law should be used only for diagram purpose for calculation of resultant vector (For addition of more than 2 vectors) we use components of vector.
KEEP IN MEMORY
- If
, then 
is a null vector. - Null vector or zero vector is defined as a vector whose magnitude is zero and direction indeterminate. Null vector differs from ordinary zero in the sense that ordinary zero is not associated with direction.
is called a unit vector. It is unit less and dimensionless vector. Its magnitude is 1. It represents direction only.- If
, then 
and 
, where 
are unit vectors of A and B respectively. - A vector can be divided or multiplied by a scalar.
- Vectors of the same kind can only be added or subtracted. It is not possible to add or subtract the vectors of different kind. This rule is also valid for scalars.
- Vectors of same as well as different kinds can be multiplied.
- A vector can have any number of components. But it can have only three rectangular components in space and two rectangular components in a plane. Rectangular components are mutually perpendicular.
- The minimum number of unequal non-coplanar whose vector sum is zero is 4.
- When
, where 
is modulus or magnitude of vector
.
makes 45° with both X and Y-axes. It makes angle 90° with Z-axis.
makes angle 54.74° with each of the X, Y and Z-axes.
- If
then angle between 
and 
is 
. - Magnitude of a vector is independent of coordinate axes system.
- Component of a vector perpendicular to itself is zero.
- Resultant of two vectors is maximum when θ = 0°, Rmax = A + B
- Resultant of two vectors is minimum when θ = 180° , Rmin = A – B
- The magnitude of resultant ofand can vary between (A + B) and (A – B)
SUBTRACTION OF VECTORS
We convert vector subtraction into vector addition.
If the angle between and is θ then the angle between
and 
is (180° – θ).
RESOLUTION OF A VECTOR
RECTANGULAR COMPONENTS OF A VECTOR IN PLANE
The vector 
may be written as
may be written aswhere 
is the component of vector 
in X-direction and 
is the component of vector in the Y-direction.
is the component of vector in X-direction and is the component of vector in the Y-direction.Also Ax = A cos θ and Ay = A sin θ
⇒ A cos θ and A sin θ are the magnitudes of the components of 
in X and Y-direction respectively.
in X and Y-direction respectively.Also 
RECTANGULAR COMPONENTS OF A VECTOR IN 3D
Three rectangular components along X, Y and Z direction are given by 
Therefore, vector may be written as
Therefore, vector may be written as
and 
If 
, 
and 
are the angles subtended by the rectangular components of vector then
, and are the angles subtended by the rectangular components of vector thencos = 
cos = 
and cos 
= 
= cos = and cos = Also, cos2 
+ cos2 
+ cos2 
= 1
+ cos2 + cos2 = 1
CAUTION
Do not resolve the vector at its head.
Do not resolve the vector at its head.
The vector is always resolved at its tail.
PRODUCT OF TWO VECTORS
SCALAR OR DOT PRODUCT
The scalar or dot product of two vectors A and B is a scalar, which is equal to the product of the magnitudes of and and cosine of the smaller angle between them.
i.e., 
= A B cosθ
= A B cosθe.g. 
PROPERTIES OF SCALAR OR DOT PRODUCT
= A (B cosθ) = B (A cosθ)
The dot product of two vectors can be interpreted as the product of the magnitude of one vector and the magnitude of the component of the other vector along the direction of the first vector.
Dot product of two vectors is commutative.
Dot product is distributive. 
= (Ax Bx + Ay By + Az Bz)
VECTOR OR CROSS PRODUCT
The vector product of two vectors is defined as a vector having magnitude equal to the product of two vectors and sine of the angle between them. Its direction is perpendicular to the plane containing the two vectors (direction of the vector is given by right hand screw rule or right hand thumb rule.
= (AB sin θ) The direction of 
perpendicular to the plane containing vectors 
and 
in the sense of advance of a right handed screw rotated from to is through the smaller angle between them.
perpendicular to the plane containing vectors and in the sense of advance of a right handed screw rotated from to is through the smaller angle between them.
e.g., 
PROPERTIES OF VECTOR OR CROSS PRODUCT
(not commutative) 
(follows distributive law)
= (Ay Bz – Az By) 
+ (Az Bx – Ax Bz) 
+ (Ax By – Ay Bx) 
+ (Az Bx – Ax Bz) + (Ax By – Ay Bx) - The cross product of two vectors represents the area of the parallelogram formed by them.
= area of parallelogram PQRS
= 2 (area of ΔPQR)
- A unit vector which is perpendicular to A as well as B is
KEEP IN MEMORY
- tan θ =
- If
, then 
- If
then angle between 
and 
is
. - If
then 
- Division by a vector is not defined. Because, it is not possible to divide by a direction.
- The sum and product of vectors is independent of coordinate axes system.
CONDITION OF ZERO RESULTANT VECTOR
If the three vectors acting on a point object at the same time are represented in magnitude and direction by the three sides of a triangle taken in order, then their resultant is zero and the three vectors are said to be in equilibrium.
i.e. 
LAMI’S THEOREM
It states that if three forces acting at a point are in equilibrium, then each force is proportional to the sine of the angle between the other two forces.
or, 
1.3.1 Distinguish between vector and scalar quantities, and give examples of each.
Scalars only have one dimension, size, while vectors have two: magnitude (size) and direction. Vectors are represented in print as bold and italicized symbols, for example: F.
| Scalars | Vectors |
|---|---|
| Mass | Force |
| Speed | Velocity |
| Charge | Acceleration |
| Distance | Displacement |
| Energy | Momentum |
When a car moves backwards, its displacement is said to be negative. Any vector with a negative value indicates that the object is moving in the opposite direction. We usually use scalar values in normal conversation, such as the distance traveled by a car, because the direction does not matter. It is important to note that if a person has walked 10m forward and 10m back again, the distance traveled was 20m but their displacement is 0 because they are in the same position. Mathematically, this would be represented as 10m forward and -10m backward, which cancel out.
1.3.2 Determine the sum or difference of two vectors by a graphical method.
Parallel vectors The sum of parallel vectors that run in the same direction can be determined by simple addition.
The sum of parallel vectors that run in the opposite direction can be determined by the subtraction of the smaller vector from the larger vector.
Vectors and scalars Multiplying vectors by scalars functions like any ordinary equation. For example, F × 2 is 2F. It follows distributive properties — 2(F + M) = 2F + 2M, and associative properties — 2(MF) = 2F(M). When graphed, vectors multiplied by scalars become longer and vectors divided by scalars become shorter. Negative vectors go in the opposite direction of their positive counterparts. Note that when a vector is multiplied by 0, it is a null vector and has no magnitude or direction.
| Adjacent vectors The sum of two vectors that are perpendicular or adjacent to each other can be determined through the Pythagoras theorem.
This can also be determined using basic trigonometry.
|
1.3.3 Resolve vectors into perpendicular components along chosen axes.
Just as two vectors can be added to form a resultant vector, a resultant vector can be split into its two components. The basic rule of thumb is:
Consider the following example:
The two vectors that run diagonally can be split into their individual components for further calculation.
It should be taken into account that the plane of reference will not always be the paper. The object may be situated at a slope.
In this case, the force of the weight is not actually running straight downwards. We need to use the slope as a reference for the components of the vector.