Magnitude of a vector and unit vectors Problem set A

Find the exact magnitude of each vector.

1. (a) \(\begin{pmatrix}2 \\ -3 \\ 6\end{pmatrix}\)
2. (b) \(\begin{pmatrix}-1 \\ 5 \\ -3\end{pmatrix}\)

Find the exact value of \(|\mathbf{a}|\) for \(\mathbf{a}=4\mathbf{i}-8\mathbf{j}+\mathbf{k}\).

Write down the two unit vectors parallel to \(\begin{pmatrix}1 \\ -2 \\ 2\end{pmatrix}\).

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

1. (a) \(4\mathbf{i}+\mathbf{j}+8\mathbf{k}\)
2. (b) \(\mathbf{i}-2\mathbf{j}\)

A drone flies in the direction of \(\begin{pmatrix}1 \\ 4 \\ 8\end{pmatrix}\) at a speed of 18 m/s. Find its velocity vector.

A tow rope pulls a barge with a force of magnitude 12 N in the direction of \(\mathbf{i}+\mathbf{j}-\mathbf{k}\). Write the force as a vector.

A wasp flies from its nest. Relative to the nest its displacement is \(\begin{pmatrix}12 \\ -5 \\ 20\end{pmatrix}\) metres, where \(\mathbf{k}\) is vertically up. How far is the wasp from the nest? Give your answer to 3 significant figures.

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

Find the two vectors of magnitude 10 that are parallel to \(\begin{pmatrix}1 \\ -2 \\ 3\end{pmatrix}\).