Addition of Vectors – Example

Vector addition can be performed using both geometric and mathematical methods.

The geometric method allows us to visualize the resultant vector, while the mathematical method provides a more analytical approach by adding the corresponding components. Both methods yield the same resultant vector.

Vector addition is the process of combining two or more vectors to obtain a resultant vector. Let’s consider two vectors $\mathbf{A}$ and $\mathbf{B}$ in a two-dimensional space. Each vector has both magnitude and direction.

 

Geometric Method

The geometric method of vector addition involves placing the tail of the second vector at the head of the first vector and drawing a new vector from the tail of the first vector to the head of the second vector.

The resultant vector, denoted as $\mathbf{R}$, is the vector that starts at the tail of the first vector and ends at the head of the second vector.

Mathematical Method

The mathematical method of vector addition involves adding the corresponding components of the vectors. If vector $\mathbf{A}$ has components $A_x$ and $A_y$, and vector $\mathbf{B}$ has components $B_x$ and $B_y$, then the components of the resultant vector $\mathbf{R}$ are given by:

\[
R_x = A_x + B_x
\] \[
R_y = A_y + B_y
\]

 

Example

Let’s consider the following vectors:

\[
\mathbf{A} = 3\mathbf{i} + 2\mathbf{j}
\] \[
\mathbf{B} = -2\mathbf{i} + 4\mathbf{j}
\]

Using the mathematical method, we can add the vectors as follows:

\[
R_x = 3 + (-2) = 1
\] \[
R_y = 2 + 4 = 6
\]

Therefore, the resultant vector $\mathbf{R}$ is given by:

\[
\mathbf{R} = 1\mathbf{i} + 6\mathbf{j}
\]