Addition of Vectors
A vector can be thought of as a displacement from point A to point B. To understand the need for the addition and subtraction of vectors, consider an example. In the figure, Laxmi goes from point A to point B and then to point C following the vectors shown in the figure. Now, after reaching point C, the net displacement made by this girl is from A to C which is given by the vector .
This can be expressed in terms of vectors as,
This is known as the triangle law of vector addition.
Triangle Law of Vector Addition
The triangle law of vector addition states that if two sides of any triangle represent the two vectors that are on a body in the same order as the side of the triangle, then the third side of the triangle represents the resultant vector.
In general, for two vectors and their addition is done such that the initial point of one vector coincides with the terminal points of the other.
In the figure, we have two vectors a and b given
Now the vector b is shifted without changing its direction and magnitude such that now the initial point of vector lies on the endpoint of vector a as shown in the figure,
In the case of subtraction, the direction of vector b is reversed, and then both vectors are added as shown in the image below,
Parallelogram Law of Vector Addition
Parallelogram law of vector addition states that if the adjacent side of the parallelogram represents two vectors then the diagonal starting from the same initial point represents the resultant of the vector.
Properties of Vector Addition
Various properties of vector addition are
Property 1: Vector addition follows commutative property. For two vectors and .
Property 2: Vector addition of three vectors follows the associative property.
Vector Algebra
Vectors algebra is the branch of algebra that involves operations on vectors. Vectors are quantities that have both magnitude and direction so normal operations are not performed on the vectors. We can add, subtract, and multiply vector quantities using special vector algebra rules. Vectors can be easily represented in 2-D or 3-D spaces. Vector algebra has various applications it is used in solving various problems in mathematics and physics, engineering, and various other fields.
In this article, we will learn about vector algebra, operations in vector algebra, types of vectors, and others in detail.