Components of a vector Problem set A

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

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

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

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

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

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

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 \(k\).

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