Components of a vector Problem set B One question at a time

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

1. (a) \(\mathbf{a}=7\mathbf{i} + \mathbf{j}\) in the direction of \(2\mathbf{i} - 3\mathbf{j}\)

   • \(\mathbf{a}\cdot\mathbf{b}=11\), \(|\mathbf{b}|=\sqrt{13}\)
   • Component \(=\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}=\dfrac{11}{\sqrt{13}}=3.05\)

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

   • \(\mathbf{a}\cdot\mathbf{b}=-34\), \(|\mathbf{b}|=\sqrt{41}\)
   • Component \(=\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}=\dfrac{-34}{\sqrt{41}}=-5.31\)

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

1. (a) \(\mathbf{a}=3\mathbf{i} + 4\mathbf{j} - 6\mathbf{k}\) in the direction of \(\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}\)

   • \(\mathbf{a}\cdot\mathbf{b}=-17\), \(|\mathbf{b}|=\sqrt{9}\)
   • Component \(=\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}=\dfrac{-17}{\sqrt{9}}=-5.67\)

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

   • \(\mathbf{a}\cdot\mathbf{b}=22\), \(|\mathbf{b}|=\sqrt{49}\)
   • Component \(=\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}=\dfrac{22}{\sqrt{49}}=3.14\)

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

1. (a) \(\mathbf{a}=\begin{pmatrix}-3\\4\end{pmatrix}\) perpendicular to \(\begin{pmatrix}5\\2\end{pmatrix}\)

   • \(|\mathbf{a}\times\mathbf{b}|=|a_1b_2-a_2b_1|=|(-3)\times2-4\times5|=26\)
   • Component \(=\dfrac{26}{|\mathbf{b}|}=\dfrac{26}{\sqrt{29}}=4.83\)

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

   • \(\mathbf{a}\times\mathbf{b}=\begin{pmatrix}(-1)\times(-1)-3\times2\\ 3\times2-4\times(-1)\\ 4\times2-(-1)\times2\end{pmatrix}=\begin{pmatrix}-5\\10\\10\end{pmatrix}\)
   • Component \(=\dfrac{|\mathbf{a}\times\mathbf{b}|}{|\mathbf{b}|}=\dfrac{\sqrt{225}}{\sqrt{9}}=5.00\)

A student pulls a sledge up a snowy slope with a force \(\mathbf{F}=\begin{pmatrix}5\\-3\\8\end{pmatrix}\) N. The direction up the slope is \(\mathbf{d}=\begin{pmatrix}2\\1\\2\end{pmatrix}\).

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

   • \(\mathbf{F}\cdot\mathbf{d}=23\), \(|\mathbf{d}|=\sqrt{9}\)
   • Component \(=\dfrac{\mathbf{F}\cdot\mathbf{d}}{|\mathbf{d}|}=\dfrac{23}{\sqrt{9}}=7.67\)

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

   • \(\mathbf{F}\times\mathbf{d}=\begin{pmatrix}(-3)\times2-8\times1\\ 8\times2-5\times2\\ 5\times1-(-3)\times2\end{pmatrix}=\begin{pmatrix}-14\\6\\11\end{pmatrix}\)
   • Component \(=\dfrac{|\mathbf{F}\times\mathbf{d}|}{|\mathbf{d}|}=\dfrac{\sqrt{353}}{\sqrt{9}}=6.26\)

A sailing boat moves in the direction \(\mathbf{d}=\begin{pmatrix}6\\2\\3\end{pmatrix}\). The wind velocity is \(\mathbf{w}=\begin{pmatrix}-3\\8\\2\end{pmatrix}\) m/s.

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

   • \(\mathbf{w}\cdot\mathbf{d}=4\), \(|\mathbf{d}|=\sqrt{49}\)
   • Component \(=\dfrac{\mathbf{w}\cdot\mathbf{d}}{|\mathbf{d}|}=\dfrac{4}{\sqrt{49}}=0.571\)
   • 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 boat's direction.

   • \(\mathbf{w}\times\mathbf{d}=\begin{pmatrix}8\times3-2\times2\\ 2\times6-(-3)\times3\\ (-3)\times2-8\times6\end{pmatrix}=\begin{pmatrix}20\\21\\-54\end{pmatrix}\)
   • Component \(=\dfrac{|\mathbf{w}\times\mathbf{d}|}{|\mathbf{d}|}=\dfrac{\sqrt{3757}}{\sqrt{49}}=8.76\)

The vector \(\mathbf{a}=\begin{pmatrix}3\\k\\2\end{pmatrix}\) has component \(5\) in the direction of \(\mathbf{b}=\begin{pmatrix}1\\4\\8\end{pmatrix}\). Find the value of \(k\).

• \(\mathbf{a}\cdot\mathbf{b}=4 k + 19\), \(|\mathbf{b}|=\sqrt{81}\)
• \(\dfrac{4 k + 19}{\sqrt{81}}=5\)
• \(4 k + 19=45\), so \(k=\frac{13}{2}\)

Take \(\mathbf{a}=\begin{pmatrix}4\\1\\-5\end{pmatrix}\) and \(\mathbf{b}=\begin{pmatrix}2\\3\\6\end{pmatrix}\). 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 \(|\mathbf{a}|^2\).

• \(\mathbf{a}\cdot\mathbf{b}=-19\), \(|\mathbf{b}|=\sqrt{49}\)
• Component \(=\dfrac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}=\dfrac{-19}{\sqrt{49}}=-2.71\)
• \(\mathbf{a}\times\mathbf{b}=\begin{pmatrix}1\times6-(-5)\times3\\ (-5)\times2-4\times6\\ 4\times3-1\times2\end{pmatrix}=\begin{pmatrix}21\\-34\\10\end{pmatrix}\)
• Component \(=\dfrac{|\mathbf{a}\times\mathbf{b}|}{|\mathbf{b}|}=\dfrac{\sqrt{1697}}{\sqrt{49}}=5.88\)
• \(\left(-2.714\right)^2+\left(5.885\right)^2=42.00\), and \(|\mathbf{a}|^2=42\)