Vector arithmetic
Multiplying vectors by a scalar
A vector describes a movement from one point to another. A vector quantity has magnitude (size) and direction. can be multiplied by a A quantity that requires only a size, for example, distance travelled is 20 m. which changes the size of the vector but not the direction.
\(\mathbf{k} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}\)
The vector \(2k\) is twice as long as the vector \(k\). Double each number in \(k\) to get \(2k\).
\(\mathbf{2k} = \begin{pmatrix} 6 \\ -4 \end{pmatrix}\)
\(\mathbf{m} = \begin{pmatrix} -4 \\ -4 \end{pmatrix}\)
The vector \(\frac{1}{2} \mathbf{m}\) is half as long as the vector \(\mathbf{m}\). Halve each number in \(\mathbf{m}\) to get \(\frac{1}{2} \mathbf{m}\).
\(\frac{1}{2} \mathbf{m} = \begin{pmatrix} -2 \\ -2 \end{pmatrix}\)
Adding vectors
Vectors can be added by drawing the first vector, then starting the second vector where the first vector ends.
\(\overrightarrow{XY} + \overrightarrow{YZ} = \overrightarrow{XZ}\)
\(\begin{pmatrix} 4 \\ 2 \end{pmatrix} + \begin{pmatrix} 1 \\ -4 \end{pmatrix} = \begin{pmatrix} 4 + 1 \\ 2 + -4 \end{pmatrix} = \begin{pmatrix} 5 \\ -2 \end{pmatrix}\)
The single vector they create (\(\overrightarrow{XZ}\)) is the The combination of two or more vectors. For example, the resultant of a 30 N force up and a 40 N force horizontally is 50 N at 37掳 above the horizontal.. vector.
Travelling from \(X\) to \(Y\), then from \(Y\) to \(Z\), is the same as travelling from \(X\) to \(Z\).
Subtracting vectors
Subtracting a vector is the same as adding a negative vector.
\(\overrightarrow{YX} + \overrightarrow{XZ} = \overrightarrow{YZ}\)
Since the vector \(\overrightarrow{YX}\) has the same magnitude but opposite direction to the vector \(\overrightarrow{XY}\):
\(\overrightarrow{YX} = \overrightarrow{-XY}\)
\(\overrightarrow{-XY} + \overrightarrow{XZ} = \overrightarrow{YZ}\)
\(- \begin{pmatrix} 4 \\ 2 \end{pmatrix} + \begin{pmatrix} 5 \\ -2 \end{pmatrix} = \begin{pmatrix} -4 + 5 \\ -2 + -2 \end{pmatrix} = \begin{pmatrix} 1 \\ -4 \end{pmatrix}\)
Example
Write, in terms of \(\mathbf{a}\) and \(\mathbf{b}\), the vector \(\overrightarrow{KM}\).
\(\overrightarrow{KO} + \overrightarrow{OM} = \overrightarrow{KM}\)
\(\overrightarrow{KM} = \mathbf{-a} + \mathbf{b}\) or \(\mathbf{b} - \mathbf{a}\)