Addition of Vectors

Vectors cannot be added by usual algebraic rules. While adding two vectors, the magnitude and the direction of the vectors must be taken into account. Triangle law is used to add two vectors, the diagram below shows two vectors “a” and “b” and the resultant calculated after their addition. Vector addition follows commutative property, this means that the resultant vector is independent of the order in which the two vectors are added. 

[Tex]\vec{a} + \vec{b} = \vec{c}[/Tex]

The commutative property of vector addition states that,

[Tex]\vec{a} + \vec{b} = \vec{b} + \vec{a}[/Tex]

Triangle Law of Vector Addition

For Triangle Law of Vector Addition, Consider the vectors given in the figure above. The line AB represents the vector “a”, and BC represents the vector “b”. The line AC represents the resultant vector. The direction of AC is from A to C.

Line AC represents, 

[Tex]\vec{a} + \vec{b}[/Tex]

The magnitude of the resultant vector is given by, 

[Tex]\sqrt{|a|^2 + |b|^2 + 2|a||b|cos(\theta)}[/Tex]

The θ represents the angle between the two vectors. Let Φ be the angle made by the resultant vector with the vector p.

[Tex]tan(\phi) = \frac{qsin(\theta)}{p + qcos(\theta)}[/Tex]

Parallelogram Law of Vector Addition

According to Parallelogram Law of Vector Addition if, “Adjacent side of a parallelogram represents two vectors then the diagonal starting from the same initial point represents the resultant of the vector.”

This is represented as by the image added below:

Here, vector A and vector B represents the sides of parallelogram PQ and QR respectively and QS represents the resultant sum vector C.

Vector Operations are operations that are performed on vector quantities. Vector quantities are the quantities that have both magnitude and direction. So performing mathematical operations on them directly is not possible. So we have special operations that work only with vector quantities and hence the name, vector operations.

Thus, It is essential to know what kind of operations can be performed on the vector quantities and vector operations tells us about the same. This article deals with vector operations, such as vector addition, the cross product of two vectors, the dot product of two vectors, and others in detail. Let’s learn about all of them in detail, here in this article.