Vector Addition & Resolution

Understanding Vectors

Vectors are quantities that have both magnitude and direction. Examples include force, velocity, and displacement.

In contrast, scalars have only magnitude, like mass or temperature.

Vector Addition

To find the resultant vector when adding two or more vectors, you can use the following methods:

• Graphical Method: Draw vectors to scale on a graph and use the head-to-tail method to find the resultant.
• Analytical Method: Use mathematical techniques such as the parallelogram law or triangle law.

Parallelogram Law

If two vectors $\vec{A}$ and $\vec{B}$ are represented as adjacent sides of a parallelogram, the diagonal of the parallelogram represents the resultant vector $\vec{R}$.

Mathematically, the magnitude of $\vec{R}$ can be found using:

$$R = \sqrt{A^2 + B^2 + 2AB \cos \theta}$$

where $\theta$ is the angle between $\vec{A}$ and $\vec{B}$.

Triangle Law

Place the tail of $\vec{B}$ at the head of $\vec{A}$. The vector from the tail of $\vec{A}$ to the head of $\vec{B}$ is the resultant vector $\vec{R}$.

Vector Resolution

Resolving a vector means breaking it down into its components, usually along the x and y axes.

For a vector $\vec{V}$ with magnitude $V$ and angle $\theta$ from the x-axis:

• Horizontal Component: $V_x = V \cos \theta$
• Vertical Component: $V_y = V \sin \theta$

Worked Example