Subtracting vectors

Subtracting a vector is the same as adding a negative version of the vector (remember that making a vector negative means reversing its direction).

\(\left( \begin{array}{l} a\\ b \end{array} \right) - \left( \begin{array}{l} c\\ d \end{array} \right) = \left( \begin{array}{l} a - c\\ b - d \end{array} \right)\)

Look at the diagram and imagine going from X to Z. How would you write the path in vectors using only the vectors \(\overrightarrow {XY}\) and \(\overrightarrow {ZY}\)?

You could say it is vector \(\overrightarrow {XY}\) followed by a backwards movement along \(\overrightarrow {ZY}\).

So we can write the path from X to Z as:

\(\overrightarrow {XY} - \overrightarrow {ZY} = \overrightarrow {XZ}\)

Written out in numbers it looks like this:

\(\left( \begin{array}{l} 4\\ 2 \end{array} \right) - \left( \begin{array}{l} 1\\ 2 \end{array} \right) = \left( \begin{array}{l} 3\\ 0 \end{array} \right)\)

Now try the example questions below.

Question

If \(x = \left( \begin{array}{l} 1\\ 3 \end{array} \right),y = \left( \begin{array}{l} - 2\\ 4 \end{array} \right)and\,z = \left( \begin{array}{l} - 1\\ - 2 \end{array} \right)\), find:

\(- y\)
\(x - y\)
\(2x + 3z\)

Question

For vectors \(u = \left( {\begin{array}{*{20}{c}} 2\\ 5\\ 9 \end{array}} \right)and\,v = \left( {\begin{array}{*{20}{c}} 7\\ 3\\ { - 4} \end{array}} \right)\) calculate, u + v.

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Using vector componentsSubtracting vectors