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

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) \(\mathbf{a}\)

   • Count across from the tail to the head: right 4
   • Then count up 3
   • \(\mathbf{a}=\begin{pmatrix}4\\3\end{pmatrix}\)

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

   • Count across from the tail to the head: right 2
   • Then count down 3
   • \(\mathbf{b}=\begin{pmatrix}2\\-3\end{pmatrix}\)

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) \(\mathbf{c}\)

   • Count across from the tail to the head: left 4
   • Then count up 2
   • \(\mathbf{c}=\begin{pmatrix}-4\\2\end{pmatrix}\)

2. (b) \(\mathbf{d}\)

   • Count across from the tail to the head: left 3
   • Then count down 5
   • \(\mathbf{d}=\begin{pmatrix}-3\\-5\end{pmatrix}\)

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

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

   • The top number goes with i and the second with j
   • \(7\mathbf{i}-2\mathbf{j}\)

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

   • The three numbers go with i, j and k
   • A zero component gives no term
   • \(-3\mathbf{i}+5\mathbf{k}\)

Write each vector as a column vector with three components.

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

   • Read the coefficients of i, j and k in that order
   • Every term is there
   • \(\begin{pmatrix}6\\-1\\3\end{pmatrix}\)

2. (b) \(5\mathbf{j}-2\mathbf{k}\)

   • Read the coefficients of i, j and k in that order
   • There is no i term, so that component is 0
   • \(\begin{pmatrix}0\\5\\-2\end{pmatrix}\)

3. (c) \(-3\mathbf{i}+4\mathbf{k}\)

   • Read the coefficients of i, j and k in that order
   • There is no j term, so that component is 0
   • \(\begin{pmatrix}-3\\0\\4\end{pmatrix}\)

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.

   • East and north are positive, west and south are negative
   • First component: east/west. Second component: north/south
   • \(\begin{pmatrix}7\\-2\end{pmatrix}\)

2. (b) Write the displacement using \(\mathbf{i}\) and \(\mathbf{j}\).

   • Coefficient of i is the first component, coefficient of j the second
   • \(7\mathbf{i}-2\mathbf{j}\)

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.

   • East, north and up are positive. West, south and down are negative
   • \(\begin{pmatrix}-30\\20\\8\end{pmatrix}\)

2. (b) Write the displacement using \(\mathbf{i}\), \(\mathbf{j}\) and \(\mathbf{k}\).

   • The three components are the coefficients of i, j and k
   • \(-30\mathbf{i}+20\mathbf{j}+8\mathbf{k}\)

3. (c) The drone's speed is 5 m/s. Is speed a vector or a scalar?

   • Speed has a size but no direction, so it is 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 \(q\).

• Write the second vector as a column: \(\begin{pmatrix}5\\4\\-3\end{pmatrix}\)
• Equal vectors have equal components: \(3p-1=5\) and \(q+2=-3\)
• \(3p=6,\ p=2\) and \(q=-5\)

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 \(x\mathbf{i}+y\mathbf{j}\).

• Count from the tail to the head: left 4, then up 5
• \(\mathbf{w}=\begin{pmatrix}-4\\5\end{pmatrix}\)
• The components are the coefficients of i and j: \(\mathbf{w}=-4\mathbf{i}+5\mathbf{j}\)