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Basic Mathematics(Pre-requisite)

Differential calculus

Do you know what a physical quantity is? A physical quantity is any physical property that can be expressed in numbers. For example, time is a physical quantity as it can be expressed in numbers, but beauty is not as it cannot be expressed in numbers.

Scalar Quantities

• If a physical quantity can be completely described only by its magnitude, then it is a scalar quantity. To measure the mass of an object, we only have to know how much matter is present in the object. Therefore, mass of an object is a physical quantity that only requires magnitude to be expressed. Therefore, we say that mass is a scalar quantity.

• Some more examples of scalar quantities are time, area, volume, and energy.

• We can add scalar quantities by simple arithmetic means.

• It is difficult to plot scalar quantities on a graph.

Vector Quantities

• There are some physical quantities that cannot be completely described only by their magnitudes. These physical quantities require direction along with magnitude. For example, if we consider force, then along with the magnitude of the force, we also have to know the direction along which the force is applied. Therefore, to describe a force, we require both its magnitude and direction. This type of physical quantity is called a vector quantity.

Therefore, we can define vector quantity as the physical quantity that requires both magnitude and direction to be described.

• Some examples of vector quantities are velocity, force, weight, and displacement.

• Vector quantities cannot be added or subtracted by simple arithmetic means.

• Vector quantities can easily be plotted on a graph.

Scalars v/s Vectors

 Scalars Vectors A scalar quantity has only magnitude. A vector quantity has both magnitude and direction. Scalars can be added, subtracted, multiplied, and divided just as ordinary numbers i.e., scalars are subjected to simple arithmetic operations. Vectors cannot be added, subtracted, and multiplied following simple arithmetic laws. Arithmetic division of vectors is not possible at all. Example: mass, volume, time, distance, speed, work, temperature Example: displacement, velocity, acceleration, force

Position Vector

Position vector of a point in a coordinate system is the straight line that joins the origin and the point.

Magnitude of the vector is the length of the straight line and its direction is along the angle θ from the positive x-axis.

Displacement Vector

Displacement vector is the straight line joining the initial and final positions.

Equality of Vectors

Two vectors  and are said to be equal, if and only if they have the same magnitude and the same direction.

Scalars vs. Vectors

 Scalars Vectors A scalar quantity has magnitude only. A vector quantity has both magnitude and direction. Scalar quantities can be added, subtracted, multiplied and divided just like ordinary numbers, i.e., scalars are subjected to simple arithmetic operations. Vectors cannot be added, subtracted or multiplied following simple arithmetic rules. Arithmetic division of vectors is not possible at all. Example: Mass, volume, time, distance, speed, work, temperature, etc. Example: Displacement, velocity, acceleration, force, etc.

Distance & Displacement:

Position Vector

The position vector of a point in a coordinate system is the straight line that joins the origin and the point.

The magnitude of a vector is the length of the straight line. Its direction is along the angle θ from the positive x-axis.

Displacement Vector

Displacement vector is the straight line joining the initial and the final position.

Equality of Vectors

Two vectors $\stackrel{\to }{A}$ and $\stackrel{\to }{B}$  are said to be equal only if they have the same magnitude and the same direction.

Negative vector
Negative vector is a vector whose magnitude is equal to that of a given vector, but whose direction is opposite to that of the given vector.

Zero vector
Zero vector is a vector whose magnitude is zero and have an arbitrary direction.

Resultant vector
The resultant vector of two or more vectors is a vector which produces the same effect as produced by the individual vectors together.

Multiplication of Vectors by Real Numbers

• Multiplication of a vector $\stackrel{\to }{A}$ with a positive number k only changes the magnitude of…

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