Magnitude of a vector and unit vectors Problem set B

Find the exact magnitude of each vector.

1. (a) \(\begin{pmatrix}6 \\ -6 \\ 7\end{pmatrix}\)
2. (b) \(\begin{pmatrix}2 \\ -4 \\ -7\end{pmatrix}\)

Find the exact value of \(|\mathbf{a}|\) for \(\mathbf{a}=12\mathbf{i}-4\mathbf{j}+3\mathbf{k}\).

Write down the two unit vectors parallel to \(\begin{pmatrix}-6 \\ 2 \\ 3\end{pmatrix}\).

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

1. (a) \(6\mathbf{i}-7\mathbf{j}+6\mathbf{k}\)
2. (b) \(3\mathbf{i}+\mathbf{j}\)

A submarine dives in the direction of \(\begin{pmatrix}2 \\ 3 \\ 6\end{pmatrix}\) at a speed of 21 m/s. Find its velocity vector.

A crane cable pulls a container with a force of magnitude 18 kN in the direction of \(\mathbf{i}-2\mathbf{j}+\mathbf{k}\). Write the force as a vector.

A hot-air balloon launches from a field. Relative to the launch point its displacement is \(\begin{pmatrix}-7 \\ 15 \\ 9\end{pmatrix}\) metres, where \(\mathbf{k}\) is vertically up. How far is the balloon from the launch point? Give your answer to 3 significant figures.

The vector \(\mathbf{u}=\begin{pmatrix}2k \\ 3k \\ 6k\end{pmatrix}\) is a unit vector. Find the possible values of \(k\).

Find the two vectors of magnitude 8 that are parallel to \(\begin{pmatrix}2 \\ 0 \\ -1\end{pmatrix}\).