Column vectors and base vectors i, j, k Problem set B

Write the vectors \(\mathbf{p}\) and \(\mathbf{q}\) as column vectors. Each grid square is one unit. The black dot is the tail and the orange dot is the head.

1. (a) \(\mathbf{p}\)
2. (b) \(\mathbf{q}\)

Write the vectors \(\mathbf{r}\) and \(\mathbf{s}\) as column vectors. Each grid square is one unit. The black dot is the tail and the orange dot is the head.

1. (a) \(\mathbf{r}\)
2. (b) \(\mathbf{s}\)

Write each column vector using the base vectors \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\).

1. (a) \(\begin{pmatrix}-6\\5\end{pmatrix}\)
2. (b) \(\begin{pmatrix}4\\-2\\0\end{pmatrix}\)

Write each vector as a column vector with three components.

1. (a) \(2\mathbf{i}-5\mathbf{j}+\mathbf{k}\)
2. (b) \(4\mathbf{j}-9\mathbf{k}\)
3. (c) \(-5\mathbf{i}+6\mathbf{k}\)

A ferry leaves a harbour and sails 9 km west and 4 km north. Let \(\mathbf{i}\) be 1 km east and \(\mathbf{j}\) be 1 km north.

1. (a) Write the displacement as a column vector.
2. (b) Write the displacement using \(\mathbf{i}\) and \(\mathbf{j}\).

A submarine leaves a dock, then moves 40 m east and 25 m south while diving 12 m. Let \(\mathbf{i}\) be 1 m east, \(\mathbf{j}\) be 1 m north and \(\mathbf{k}\) be 1 m upwards.

1. (a) Write the displacement as a column vector.
2. (b) Write the displacement using \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\).
3. (c) The submarine's velocity is 3 m/s towards the north. Is velocity a vector or a scalar?

The vectors \(\begin{pmatrix}3p-2\\6\\3-q\end{pmatrix}\) and \(6\mathbf{j}+4\mathbf{k}-11\mathbf{i}\) are equal. Find the values of \(p\) and \(q\).

The vector \(\mathbf{v}\) is drawn on the grid. Each grid square is one unit. The black dot is the tail and the orange dot is the head. Write \(\mathbf{v}\) as a column vector and in the form \(x\mathbf{i}+y\mathbf{j}\).