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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} = \begin{pmatrix} v_x \\ v_y \end{pmatrix}

\]

where \( v_x \) and \( v_y \) are the components of the vector in each respective direction.

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}

\]

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} = \begin{pmatrix} 1 \ 0 \end{pmatrix}, \quad \mathbf{j} = \begin{pmatrix} 0 \ 1 \end{pmatrix}

\]

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

\[

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

\]

For example, the vector \( \mathbf{v} = \begin{pmatrix} 3 \ 4 \end{pmatrix} \) can be written as:

\[

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

\]

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

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

\[

\mathbf{u} + \mathbf{v} = \begin{pmatrix} u_x \\ u_y \end{pmatrix} + \begin{pmatrix} v_x \\ v_y \end{pmatrix} = \begin{pmatrix} u_x + v_x \\ u_y + v_y \end{pmatrix}

\]

Similarly, subtraction works component-wise:

\[

\mathbf{u} - \mathbf{v} = \begin{pmatrix} u_x \\ u_y \end{pmatrix} - \begin{pmatrix} v_x \\ v_y \end{pmatrix} = \begin{pmatrix} u_x - v_x \\ u_y - v_y \end{pmatrix}

\]

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 \begin{pmatrix} v_x \ v_y \end{pmatrix} = \begin{pmatrix} k v_x \ k v_y \end{pmatrix}

\]

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