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

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

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

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

1. (a) \(\mathbf{c}\)
2. (b) \(\mathbf{d}\)

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

1. (a) \(\begin{pmatrix}7\\-2\end{pmatrix}\)
2. (b) \(\begin{pmatrix}-3\\0\\5\end{pmatrix}\)

Write each vector as a column vector with three components.

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

A hiker leaves a mountain hut and walks 7 km east and 2 km south. 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 drone takes off, then flies 30 m west and 20 m north while climbing 8 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 drone's speed is 5 m/s. Is speed a vector or a scalar?

The vectors \(\begin{pmatrix}3p-1\\4\\q+2\end{pmatrix}\) and \(5\mathbf{i}+4\mathbf{j}-3\mathbf{k}\) are equal. Find the values of \(p\) and \(q\).

The vector \(\mathbf{w}\) 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 \(\mathbf{w}\) as a column vector and in the form \(x\mathbf{i}+y\mathbf{j}\).