A vector is typically represented as a line segment. The direction of the vector can be thought of as a displacement going from the tail to the head. The arrow indicates the direction of travel. The vector is usually labeled by a lower case bold letter.
Two or more vectors are identical if they have the same direction and magnitude, their tails and heads need not be in the same position in space.
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