Components of a vector Problem set A Worked solutions

Find the component of the first vector in the direction of the second.

1. (a) \(\mathbf{a}=3\mathbf{i} - 5\mathbf{j}\) in the direction of \(6\mathbf{i} + 2\mathbf{j}\)

   • \(\mathbf{a}\cdot\mathbf{b}=8\), \(|\mathbf{b}|=\sqrt{40}\)
   • Component \(=\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}=\dfrac{8}{\sqrt{40}}=1.26\)

2. (b) \(\mathbf{a}=\begin{pmatrix}-4\\3\end{pmatrix}\) in the direction of \(\begin{pmatrix}2\\-1\end{pmatrix}\)

   • \(\mathbf{a}\cdot\mathbf{b}=-11\), \(|\mathbf{b}|=\sqrt{5}\)
   • Component \(=\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}=\dfrac{-11}{\sqrt{5}}=-4.92\)

Find the component of the first vector in the direction of the second.

1. (a) \(\mathbf{a}=5\mathbf{i} - \mathbf{j} + 4\mathbf{k}\) in the direction of \(2\mathbf{i} + \mathbf{j} - 2\mathbf{k}\)

   • \(\mathbf{a}\cdot\mathbf{b}=1\), \(|\mathbf{b}|=\sqrt{9}\)
   • Component \(=\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}=\dfrac{1}{\sqrt{9}}=0.333\)

2. (b) \(\mathbf{a}=\begin{pmatrix}1\\6\\-3\end{pmatrix}\) in the direction of \(\begin{pmatrix}4\\-3\\5\end{pmatrix}\)

   • \(\mathbf{a}\cdot\mathbf{b}=-29\), \(|\mathbf{b}|=\sqrt{50}\)
   • Component \(=\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}=\dfrac{-29}{\sqrt{50}}=-4.10\)

Find the component of the first vector perpendicular to the second.

1. (a) \(\mathbf{a}=\begin{pmatrix}5\\2\end{pmatrix}\) perpendicular to \(\begin{pmatrix}1\\3\end{pmatrix}\)

   • \(|\mathbf{a}\times\mathbf{b}|=|a_1b_2-a_2b_1|=|5\times3-2\times1|=13\)
   • Component \(=\dfrac{13}{|\mathbf{b}|}=\dfrac{13}{\sqrt{10}}=4.11\)

2. (b) \(\mathbf{a}=\begin{pmatrix}2\\1\\4\end{pmatrix}\) perpendicular to \(\begin{pmatrix}3\\-2\\1\end{pmatrix}\)

   • \(\mathbf{a}\times\mathbf{b}=\begin{pmatrix}1\times1-4\times(-2)\\ 4\times3-2\times1\\ 2\times(-2)-1\times3\end{pmatrix}=\begin{pmatrix}9\\10\\-7\end{pmatrix}\)
   • Component \(=\dfrac{|\mathbf{a}\times\mathbf{b}|}{|\mathbf{b}|}=\dfrac{\sqrt{230}}{\sqrt{14}}=4.05\)

A winch pulls a crate up a ramp with a force \(\mathbf{F}=\begin{pmatrix}7\\2\\-4\end{pmatrix}\) kN. The direction up the ramp is \(\mathbf{d}=\begin{pmatrix}3\\6\\2\end{pmatrix}\).

1. (a) Find the component of the force along the ramp.

   • \(\mathbf{F}\cdot\mathbf{d}=25\), \(|\mathbf{d}|=\sqrt{49}\)
   • Component \(=\dfrac{\mathbf{F}\cdot\mathbf{d}}{|\mathbf{d}|}=\dfrac{25}{\sqrt{49}}=3.57\)

2. (b) Find the component of the force perpendicular to the ramp.

   • \(\mathbf{F}\times\mathbf{d}=\begin{pmatrix}2\times2-(-4)\times6\\ (-4)\times3-7\times2\\ 7\times6-2\times3\end{pmatrix}=\begin{pmatrix}28\\-26\\36\end{pmatrix}\)
   • Component \(=\dfrac{|\mathbf{F}\times\mathbf{d}|}{|\mathbf{d}|}=\dfrac{\sqrt{2756}}{\sqrt{49}}=7.50\)

A drone flies in the direction \(\mathbf{d}=\begin{pmatrix}2\\3\\6\end{pmatrix}\). The wind velocity is \(\mathbf{w}=\begin{pmatrix}9\\-4\\1\end{pmatrix}\) m/s.

1. (a) Find the component of the wind in the direction the drone is flying. Is the wind helping or opposing the drone?

   • \(\mathbf{w}\cdot\mathbf{d}=12\), \(|\mathbf{d}|=\sqrt{49}\)
   • Component \(=\dfrac{\mathbf{w}\cdot\mathbf{d}}{|\mathbf{d}|}=\dfrac{12}{\sqrt{49}}=1.71\)
   • The component is positive, so the wind has a part in the direction of travel. It helps.

2. (b) Find the component of the wind perpendicular to the drone's direction.

   • \(\mathbf{w}\times\mathbf{d}=\begin{pmatrix}(-4)\times6-1\times3\\ 1\times2-9\times6\\ 9\times3-(-4)\times2\end{pmatrix}=\begin{pmatrix}-27\\-52\\35\end{pmatrix}\)
   • Component \(=\dfrac{|\mathbf{w}\times\mathbf{d}|}{|\mathbf{d}|}=\dfrac{\sqrt{4658}}{\sqrt{49}}=9.75\)

The vector \(\mathbf{a}=\begin{pmatrix}k\\2\\-1\end{pmatrix}\) has component \(4\) in the direction of \(\mathbf{b}=\begin{pmatrix}2\\3\\6\end{pmatrix}\). Find the value of \(k\).

• \(\mathbf{a}\cdot\mathbf{b}=2 k\), \(|\mathbf{b}|=\sqrt{49}\)
• \(\dfrac{2 k}{\sqrt{49}}=4\)
• \(2 k=28\), so \(k=14\)

Take \(\mathbf{a}=\begin{pmatrix}6\\-3\\2\end{pmatrix}\) and \(\mathbf{b}=\begin{pmatrix}1\\2\\2\end{pmatrix}\). Find the component of \(\mathbf{a}\) in the direction of \(\mathbf{b}\) and perpendicular to \(\mathbf{b}\). Show that the squares of the two components add up to \(|\mathbf{a}|^2\).

• \(\mathbf{a}\cdot\mathbf{b}=4\), \(|\mathbf{b}|=\sqrt{9}\)
• Component \(=\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}=\dfrac{4}{\sqrt{9}}=1.33\)
• \(\mathbf{a}\times\mathbf{b}=\begin{pmatrix}(-3)\times2-2\times2\\ 2\times1-6\times2\\ 6\times2-(-3)\times1\end{pmatrix}=\begin{pmatrix}-10\\-10\\15\end{pmatrix}\)
• Component \(=\dfrac{|\mathbf{a}\times\mathbf{b}|}{|\mathbf{b}|}=\dfrac{\sqrt{425}}{\sqrt{9}}=6.87\)
• \(\left(1.333\right)^2+\left(6.872\right)^2=49.00\), and \(|\mathbf{a}|^2=49\)