Introduction to Vectors

Basics of Vectors

• A vector has both magnitude and direction, but a scalar has magnitude only. For example, \(35\) metres due north, displacements, and velocity.

• A scalar quantity only contains magnitude. For example, distance, speed, mass and time.

• A vector may be represented by \(\overrightarrow{OA}\), \(\underset{\sim}{a}\), \(\mathbf{a}\) or \(\begin{pmatrix} p \\ q \end{pmatrix}\).

Note that \(\underset{\sim}{a} \neq a\) when we write. \(\underset{\sim}{a}\) is a vector while \(a\) is a scalar.

• \(\begin{pmatrix} p \\ q \end{pmatrix}\) is called the column vector, where \(p\) and \(q\) are components in the \(x\)- and \(y\)-directions respectively.

Magnitude of a Vector

• The magnitude of a vector is the length of the corresponding line segment. The magnitude of \(AB\), denoted by \(\left| \overrightarrow{AB} \right|\).

• The magnitude of a column vector \(\mathbf{a} = \begin{pmatrix} p \\ q \end{pmatrix}\) is given by \(|\mathbf{a}| = \sqrt{p^2 + q^2}\).

Write down the column vectors and magnitudes of \(\overrightarrow{AB}\), \(\overrightarrow{PQ}\) and \(\mathbf{c}\) in the \(1\) unit by \(1\) unit square grid below.