A two dimensional vector can be represented by a column vector. The top number is the x displacement and the bottom number is the y displacment:
\( \mathbf{a}= \begin{bmatrix} 1 \\ 3 \end{bmatrix} \)
A vector can be multiplied by a scalar quantity, where the x and y components are multiplied by a constant. The new vector has the same direction as shown where:
\( \mathbf{b}= 2 \mathbf{a}= \begin{bmatrix} 2 \\ 6 \end{bmatrix} \)
The vector changes direction when multiplied by a negative number.
\( \mathbf{-a}= \begin{bmatrix} -1 \\ -3 \end{bmatrix} \)
For the vector \( \mathbf{a}= \begin{bmatrix} x \\ y \end{bmatrix} \) the magnitude can be calculated using the Pythagoras theorem \( \lvert \mathbf{a} \rvert =\sqrt{x^2+y^2} \).
For example when: \( \mathbf{a}= \begin{bmatrix} 1 \\ 3 \end{bmatrix} \)
\( \lvert \mathbf{a} \rvert =\sqrt{1^2+3^2} \) simplifying we get \( \lvert \mathbf{a} \rvert =\sqrt{10} \)
The unit vector which has the same as direction as \( \mathbf{a} \) is found by dividing \( \mathbf{a} \) by it's modulus , and this vector has a magnitude of 1