Scalar and Vector Quantities‌

Scalar quantity

The quantity with only magnitude and no direction is known as a scalar quantity. Some physical quantities have only magnitude like mass, time, distance, speed, area, volume, temperature, density, etc. These quantities do not tell any direction.

Vector quantity

The physical quantities that have both direction and magnitude are known as vector quantities. The vector with a unit magnitude and direction is known as a unit vector. It is represented in lower cases with a hat circumflex. Quantities like acceleration, displacement, force, linear momentum, angular velocity, electric field, polarization, etc., are all vector quantities.

Laws of addition of vectors

1. Triangle law: - If the two sides of a triangle denote two vectors, then the third leg of the triangle denotes the resultant of these vectors.

2. Parallelogram law: - If the two adjacent sides of a parallelogram denote two vectors, then the diagonal of the parallelogram denotes the resultant of these vectors.

Difference between scalar and vector quantities

Scalar	Vector
It has only magnitude, no direction.	It has both magnitude and direction.
It is one-dimensional.	It can be one, two or three dimensional.
Any change in the scalar quantity, reflects a change in its magnitude.	Any change in the vector quantity can reflect a change in its magnitude or direction.
It cannot be resolved as it has only magnitude.	It can be resolved using sine and cosine trigonometric functions to find the resultant.
Any mathematical operation on two scalar quantities will result in a scalar quantity. If a scalar and vector quantity is operated, then the result will be a vector quantity.	Any mathematical operation on two or more vectors can result in either a scalar or a vector quantity. The dot product on two vectors gives a scalar quantity, while the cross product gives a vector quantity.
e.g. length, height, weight, mass, volume, etc.	e.g. displacement, velocity, acceleration, force, etc.

Dot product of two vectors

The dot product of two vectors is given by a · b = |a| × |b| × cos (θ), where |a| is the magnitude (length) of vector a, |b| is the magnitude (length) of vector b, θ is the angle between a and b.

Example

Find the dot product of the given figure.

Solution

We know, a · b = |a| × |b| × cos(θ)

From the figure, we can see two vectors are 10 and 13.

Therefore, a · b = 10 × 13 × cos(59.5°)

a · b = 10 × 13 × 0.5075...

a · b = 65.98

Other way to calculate-

a · b = ax × bx + ay × by

a · b = -6 × 5 + 8 × 12 = -30 + 96 = 66

Cross product of two vectors

The cross product of two vectors is given by a × b = |a| |b| sin(θ) n, where |a| is the magnitude (length) of vector a, |b| is the magnitude (length) of vector b, θ is the angle between a and b, and n is the unit vector at right angles to both a and b.

Example

Find the cross product of a = (2,3,4) and b= (5,6,7)

Solution

c_x = aybz − azby = 3×7 − 4×6 = −3

c_y = azbx − axbz = 4×5 − 2×7 = 6

c_z = axby − aybx = 2×6 − 3×5 = −3

Therefore, a × b = (−3,6,−3)