Using Vectors

A vector is a quantity that has both magnitude (size) and direction, unlike a scalar, which only has magnitude. Vectors are often used to represent physical quantities such as displacement, velocity, force, and acceleration.

Vectors can be written in component form as:

$$\mathbf{v}=\left(\begin{array}{c}{v}_x\\ v_y\end{array}\right)$$

where $v_x$ and $v_y$ are the components of the vector in each respective direction.

Magnitude of a Vector

The magnitude (or length) of a vector $\mathbf{v}$ is a scalar that represents the size of the vector. It can be calculated using Pythagoras' theorem. For a 2D vector:

$$|\mathbf{v}|=\sqrt{v_x^2+v_y^2}$$

$\mathbf{i}$ and $\mathbf{j}$ Notation

Vectors can also be expressed using the unit vectors $\mathbf{i}$ and $\mathbf{j}$, which represent the directions along the x-axis and y-axis, respectively:

$$\mathbf{i}=\left(\begin{array}{c}10\end{array}\right),\mathbf{j}=\left(\begin{array}{c}01\end{array}\right)$$

Therefore, any vector $\mathbf{v}$ can be written as:

$$\mathbf{v}=v_x\mathbf{i}+v_y\mathbf{j}$$

For example, the vector $\mathbf{v}=\left(\begin{array}{c}34\end{array}\right)$ can be written as:

$$\mathbf{v}=3\mathbf{i}+4\mathbf{j}$$

This is especially useful when working with directions in a coordinate plane.

Adding and Subtracting Vectors

Vectors can be added or subtracted by adding or subtracting their components:

$$\mathbf{u}+\mathbf{v}=\left(\begin{array}{c}{u}_x\\ u_y\end{array}\right)+\left(\begin{array}{c}{v}_x\\ v_y\end{array}\right)=\left(\begin{array}{c}{u}_x+v_x\\ u_y+v_y\end{array}\right)$$

Similarly, subtraction works component-wise:

$$\mathbf{u}-\mathbf{v}=\left(\begin{array}{c}{u}_x\\ u_y\end{array}\right)-\left(\begin{array}{c}{v}_x\\ v_y\end{array}\right)=\left(\begin{array}{c}{u}_x-v_x\\ u_y-v_y\end{array}\right)$$

Multiplying a Vector by a Scalar

When a vector is multiplied by a scalar ( k ), the magnitude of the vector is scaled, but its direction remains unchanged (unless the scalar is negative, which reverses the direction):

$$k\mathbf{v}=k\left(\begin{array}{c}{v}_x{v}_y\end{array}\right)=\left(\begin{array}{c}k{v}_{x}k{v}_y\end{array}\right)$$