Components of a vector Problem set B

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

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

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

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

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

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

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 \(k\).

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 \(|\mathbf{a}|^2\).