Magnitude of a vector and unit vectors Problem set A One question at a time

Find the exact magnitude of each vector.

1. (a) \(\begin{pmatrix}2 \\ -3 \\ 6\end{pmatrix}\)

   • \(|\mathbf{a}|=\sqrt{2^2+(-3)^2+6^2}=\sqrt{49}=7\)

2. (b) \(\begin{pmatrix}-1 \\ 5 \\ -3\end{pmatrix}\)

   • \(|\mathbf{a}|=\sqrt{(-1)^2+5^2+(-3)^2}=\sqrt{35}\)

Find the exact value of \(|\mathbf{a}|\) for \(\mathbf{a}=4\mathbf{i}-8\mathbf{j}+\mathbf{k}\).

• \(|\mathbf{a}|=\sqrt{4^2+(-8)^2+1^2}=\sqrt{81}=9\)
• Square each coefficient: \(4^2\), \((-8)^2\), \(1^2\).

Write down the two unit vectors parallel to \(\begin{pmatrix}1 \\ -2 \\ 2\end{pmatrix}\).

• \(|\mathbf{a}|=\sqrt{1^2+(-2)^2+2^2}=\sqrt{9}=3\)
• \(\frac{1}{3}\begin{pmatrix}1 \\ -2 \\ 2\end{pmatrix}=\begin{pmatrix}\frac{1}{3} \\ - \frac{2}{3} \\ \frac{2}{3}\end{pmatrix}\)
• The other one is the negative: \(\begin{pmatrix}- \frac{1}{3} \\ \frac{2}{3} \\ - \frac{2}{3}\end{pmatrix}\).

Find a unit vector in the same direction as each vector.

1. (a) \(4\mathbf{i}+\mathbf{j}+8\mathbf{k}\)

   • \(|\mathbf{a}|=\sqrt{4^2+1^2+8^2}=\sqrt{81}=9\)
   • Divide each coefficient by \(9\).

2. (b) \(\mathbf{i}-2\mathbf{j}\)

   • \(|\mathbf{a}|=\sqrt{1^2+(-2)^2+0^2}=\sqrt{5}\)
   • Divide each coefficient by \(\sqrt{5}\).

A drone flies in the direction of \(\begin{pmatrix}1 \\ 4 \\ 8\end{pmatrix}\) at a speed of 18 m/s. Find its velocity vector.

• \(|\mathbf{d}|=\sqrt{1^2+4^2+8^2}=\sqrt{81}=9\)
• \(18\times\frac{1}{9}\begin{pmatrix}1 \\ 4 \\ 8\end{pmatrix}=\begin{pmatrix}2 \\ 8 \\ 16\end{pmatrix}\)
• Check: the magnitude of the answer is \(18\).

A tow rope pulls a barge with a force of magnitude 12 N in the direction of \(\mathbf{i}+\mathbf{j}-\mathbf{k}\). Write the force as a vector.

• \(|\mathbf{d}|=\sqrt{1^2+1^2+(-1)^2}=\sqrt{3}\)
• \(\frac{12}{\sqrt{3}}\begin{pmatrix}1 \\ 1 \\ -1\end{pmatrix}\)
• \(=\begin{pmatrix}4 \sqrt{3} \\ 4 \sqrt{3} \\ - 4 \sqrt{3}\end{pmatrix}\)

A wasp flies from its nest. Relative to the nest its displacement is \(\begin{pmatrix}12 \\ -5 \\ 20\end{pmatrix}\) metres, where \(\mathbf{k}\) is vertically up. How far is the wasp from the nest? Give your answer to 3 significant figures.

• \(|\mathbf{d}|=\sqrt{12^2+(-5)^2+20^2}=\sqrt{569}=23.854\ldots\)
• To 3 significant figures: 23.9 m.

The vector \(\mathbf{u}=\begin{pmatrix}k \\ 2k \\ -2k\end{pmatrix}\) is a unit vector. Find the possible values of \(k\).

• \(|\mathbf{u}|=\sqrt{k^2+4k^2+4k^2}=3|k|\)
• A unit vector has magnitude 1, so \(3|k|=1\).
• \(k=\pm\frac{1}{3}\)

Find the two vectors of magnitude 10 that are parallel to \(\begin{pmatrix}1 \\ -2 \\ 3\end{pmatrix}\).

• \(|\mathbf{a}|=\sqrt{1^2+(-2)^2+3^2}=\sqrt{14}\)
• \(10\times\frac{1}{\sqrt{14}}\begin{pmatrix}1 \\ -2 \\ 3\end{pmatrix}=\begin{pmatrix}\frac{5 \sqrt{14}}{7} \\ - \frac{10 \sqrt{14}}{7} \\ \frac{15 \sqrt{14}}{7}\end{pmatrix}\)
• The other vector is the negative: \(\begin{pmatrix}- \frac{5 \sqrt{14}}{7} \\ \frac{10 \sqrt{14}}{7} \\ - \frac{15 \sqrt{14}}{7}\end{pmatrix}\).