Vector Operations

Vectors are physical quantities that have both magnitude and direction. The vector operations include addition, subtraction, and scalar multiplication.

A vector multiplied by a scalar value is called scalar multiplication. The result obtained is a new vector with a different magnitude. If the scalar is positive, the direction of the vector remains the same, but if it is negative, the direction of the vector is reversed. For example, the product of the mass and velocity yields the momentum. Here, the scalar quantity mass is multiplied by the vector quantity velocity, which results in the vector quantity momentum. The direction of the resultant momentum vector is in the same direction as the velocity vector.

The two vectors can be added geometrically by using the parallelogram law of vector addition. When added using this law, the two vectors are represented as adjacent sides of a parallelogram. The tails of both vectors are placed at the origin, and the lines parallel to each vector are drawn to form a parallelogram. The parallelogram's diagonal is then drawn from the origin to the opposite corner of the parallelogram to represent the resultant vector. The magnitude and direction of the resultant vector can be determined by measuring the length and angles of the parallelogram's diagonal. The parallelogram law can be simplified in a special scenario where the two vectors are collinear, meaning they have the same line of action. In this case, the resultant vector can be obtained through an algebraic or scalar addition of the two vectors.

Another method of graphically adding two vectors is the triangle law of vector addition. The triangle rule of vector addition states that if two vectors are arranged joined head to tail as two sides of a triangle, the third side of the triangle represents the magnitude and direction of the resultant vector. Vector addition using this law involves placing the first vector at the origin and placing the second vector head to tail with the first vector. The third side of the triangle from the origin to the end of the second vector represents the resultant vector.

The process of vector subtraction involves adding the reverse (negative) of one vector to another vector. This means that the rules of vector addition can be applied to obtain the difference between the two vectors.