\(\mathbf{p}=\binom{4}{-3}\) and \(\mathbf{q}=\binom{-2}{1}\). The vector \(\mathbf{p}+k\mathbf{q}\) is parallel to \(\binom{1}{-1}\). Find the value of \(k...
Question 1 Report
\(\mathbf{p}=\binom{4}{-3}\) and \(\mathbf{q}=\binom{-2}{1}\).
The vector \(\mathbf{p}+k\mathbf{q}\) is parallel to \(\binom{1}{-1}\).
Find the value of \(k\).
Answer Details
A vector \(\binom{X}{Y}\) is parallel to \(\binom{1}{-1}\) when it is a scalar multiple of it, which means \(Y=-X\): the vertical component must be the negative of the horizontal one.
- Build the combination: \(\mathbf{p}+k\mathbf{q}=\binom{4}{-3}+k\binom{-2}{1}=\binom{4-2k}{-3+k}\) [M1]
- Impose the parallel condition: \(-3+k=-(4-2k)\) [M1] oe
- Expand the right-hand side: \(-3+k=-4+2k\) [M1]
- Rearranging, \(-3+4=2k-k\), so \(k=1\) [A1]
Check the result: with \(k=1\), \(\mathbf{p}+\mathbf{q}=\binom{2}{-2}=2\binom{1}{-1}\), which is indeed parallel. The condition to state is that the components are in the ratio \(1:-1\); setting the two components equal to each other instead of opposite is the usual error, since \(\binom{1}{-1}\) points down and to the right.
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