1. Introduction & Representation
A vector is a quantity with both magnitude and direction. It is represented as a directed line segment \(\vec{AB}\) or as a column vector \(\vec{a} = \begin{pmatrix} x \\ y \end{pmatrix}\).
• Position Vector: \(\vec{OP} = \begin{pmatrix} x \\ y \end{pmatrix}\)
• Vector between points \(A(x_1, y_1)\) and \(B(x_2, y_2)\):
\(\vec{AB} = \begin{pmatrix} x_2 - x_1 \\ y_2 - y_1 \end{pmatrix}\)
2. Magnitude & Direction
The magnitude is the length of the vector, and the direction is the angle \(\theta\) relative to the positive x-axis.
• Magnitude: \(|\vec{a}| = \sqrt{x^2 + y^2}\)
• Direction Angle: \(\theta = \tan^{-1}\left(\frac{y}{x}\right)\)
• Unit Vector: \(\hat{a} = \frac{\vec{a}}{|\vec{a}|}\)
3. Vector Operations
• Addition: \(\begin{pmatrix} x_1 \\ y_1 \end{pmatrix} + \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 + x_2 \\ y_1 + y_2 \end{pmatrix}\)
• Subtraction: \(\begin{pmatrix} x_1 \\ y_1 \end{pmatrix} - \begin{pmatrix} x_2 \\ y_2 \end{pmatrix} = \begin{pmatrix} x_1 - x_2 \\ y_1 - y_2 \end{pmatrix}\)
• Scalar Multiplication: \(k \begin{pmatrix} x \\ y \end{pmatrix} = \begin{pmatrix} kx \\ ky \end{pmatrix}\)