In Physics, vector quantities are quantities that have a magnitude and direction. It is important to understand how operations like addition and subtraction are carried out on vectors. In this chapter, we will learn about these quantities and their addition and subtraction operations. Let us begin with the addition of vectors.
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Scalar and Vector Quantities
- Scalar Quantities: The physical quantities which are specified with the magnitude or size alone are referred to as Scalar Quantities. It is one – dimensional measurement of quantity like mass or temperature. A physical quantity which has force but no direction. For example, length, speed, work, mass, density, etc.
- Vector Quantities: Vector Quantities refers to those physical quantities which are characterized by the presence of both magnitudes as well as direction. It is indeed a quantity that requires both of magnitude and direction in order to identify the right quantity. For example, displacement, force, torque, momentum, acceleration, velocity, etc.
Browse more Topics under Motion In A Plane
- Introduction to Motion in a Plane
- Scalars and Vectors
- Resolution of Vectors and Vector Addition
- Relative Velocity in Two Dimensions
- Uniform Circular Motion
- Projectile Motion
Characteristics of Vectors
The characteristics of vectors are as follows
- They possess both magnitudes as well as direction.
- They do not obey the ordinary laws of Algebra.
- These change if either the magnitude or direction change or both change.
- Vectors are significantly represented by the bold-faced letters or letter which have an arrow over them.
Types of Vectors
Unit Vector
A unit vector is that vector which is a vector of unit magnitude and points in a particular direction. It is specifically used for the direction only. The unit vector is represented by putting a cap (^) over the specified quantity. The unit vector in the direction of \( \vec{A} \) is denoted by \( \hat{A} \) and is defined by,
| A | \( \hat{A} \) = \( \vec{A} \)
The unit vectors along the x, y and z-axis can be denoted as \( \hat{i} \), \( \hat{j} \), and \( \hat{k} \) respectively.
Equal Vectors
Vectors A and B are said to be equal if |A| = |B| as well as their directions, are same.
Zero Vectors
Zero vector is a vector with zero magnitudes and an arbitrary direction is called a zero vector. It can be represented by O and is also known as Null Vector.
Negative of a Vector
The vector whose magnitude is same as that of a (vector) but the direction is opposite to that of a (vector) is referred to as the negative of a (vector) and is written as – a (vector).
Addition of Vectors
As per the geometrical method for the addition of vectors, two vectors a and b. By drawing them to a common scale and placing them according to head to tail, it may be added geometrically. The vector that gets connected to the tail of the first to the head of the second is the sum of vector c. The vector addition obeys the law of associativity and is commutative.
Analytical Method i.e. Parallelogram Law for Addition of Vectors
If the two vector a and b are given such that the angle between them is θ, in that case, the magnitude of the resultant vector c of the addition of vectors is stated by –
c = √ (a^{2} + b^{2} + 2abcos θ)
And its direction is given by an angle Φ with vector a,
Tan Φ = b sin θ / (a + b cos θ)
Subtraction of Vectors
In order to subtract vector b from a, the direction must be reverse of vector b to get vector (-b). Then it must be added : (-b) to a.
Vectors |
Addition Vectors |
Subtraction of Vectors |
A = A_{x} î +A_{y} ĵ
and B = B_{x} î +B_{y} ĵ |
R = A + B
R = R_{x} î + R_{y} ĵ where R_{x} = A_{x} + B_{x} and R_{y} = A_{y} + B_{y} |
R = A – B
R = R_{x} î – R_{y} ĵ where R_{x} = A_{x} – B_{x} and R_{y} = A_{y} – B_{y} |
A = A_{x} î +A_{y} ĵ+A_{z} k̂
and B = B_{x} î +B_{y} ĵ+B_{z }k̂ |
R = A + B
R = R_{x}î + Ryĵ + Rzk̂ where R_{x} = A_{x} + B_{x} and R_{y} = A_{y} + B_{y} and R_{z} = A_{z} – B_{z} |
R = A – B
R = Rxî + R_{y} ĵ + R_{z }k̂ where R_{x} = A_{x} – B_{x} and R_{y} = A_{y} – B_{y} and R_{z} = A_{z} – B_{z} |
Solution Examples for You
Question: Read each statement below carefully and state, with reasons and examples, if it is true or false :
A scalar quantity is one that –
- is conserved in a process
- can never take negative values
- has the same value for observers with different orientations of axes
- must be dimensionless
- does not vary from one point to another in space
Solution:
- False: Despite being a scalar quantity, energy is not conserved in inelastic collisions.
- False: Despite being a scalar quantity, the temperature can take negative values.
- True: The value of a scalar does not vary for observers with different orientations of axes.
- False: Total path length is a scalar quantity. Yet it has the dimension of length.
- False: A scalar quantity like that of a gravitational potential can vary from one point to another in space.
Question: If a = 2i + j and b = 4i + 7j
- Find the components of vector c = a + b
- Find the magnitude of c and its angle with x – axis.
Solution:
- c = a + b = (2i + j) + (4i + 7j) ⇒ C = (2 + 4) i + (1 + 7) j ⇒ Thus, cx = 6 and cy = 8
- c = √ cx2 + cy2 = √ (6)2 + (8)2 = 100 ⇒ tan θ = cy/cx = 8/6 = 4/3 ⇒ θ = tan -1 (4/3) = 53 degree
which situation we use vectors subtraction?