VECTORS
CONTENT
(a) Vectors as directed line segment.
(b) Cartesian components of a vector.
(c) Magnitude of a vector, Equal vectors, Addition and subtraction of vectors, zero vectors, parallel vectors, multiplication of a vector by a scalar.
Vectors as directed line segment
A vector is any quantity which has direction as well as magnitude or size. Displacement, velocity, force, acceleration are all examples of vectors.
Since the points are on a Cartesian plane, AB can also be written as a column matrix, or column vector:
\(AB = a = \begin{pmatrix}4 \\ 2\end{pmatrix}\), Direction is important. BA is in the opposite direction to AB, although they are both parallel and have the same size: \(BA = -AB = -\begin{pmatrix}4 \\ 2\end{pmatrix} = \begin{pmatrix}-4 \\ -2\end{pmatrix}\)
A displacement vector is a movement in a certain direction without turning.
The vector ‘a’ is called the position vector of AB
Hence if a point has coordinates \((x , y)\), its position vector is \(\begin{pmatrix}x \\ y\end{pmatrix}\)
The figure shows the position vectors \(\overrightarrow{OA},\overrightarrow{OB}, \overrightarrow{OC}, \overrightarrow{OD} \)
In the figure above, the position vectors are as follows:
\(\overrightarrow{OA} = \begin{pmatrix}3 \\ 5 \end{pmatrix}, \overrightarrow{OB} = \begin{pmatrix}5 \\ -2 \end{pmatrix}, \overrightarrow{OC} = \begin{pmatrix}-3 \\ -4 \end{pmatrix}, \overrightarrow{OD} = \begin{pmatrix}-1 \\ 3 \end{pmatrix}\)
Class Activity:
Draw line segments to represent the following vectors.
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