Column vectors and base vectors i, j, k Problem set B One question at a time

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) \(\mathbf{p}\)

   • Count across from the tail to the head: right 5
   • Then count up 2
   • \(\mathbf{p}=\begin{pmatrix}5\\2\end{pmatrix}\)

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

   • Count across from the tail to the head: right 1
   • Then count down 4
   • \(\mathbf{q}=\begin{pmatrix}1\\-4\end{pmatrix}\)

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) \(\mathbf{r}\)

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

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

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

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

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

   • The top number goes with i and the second with j
   • \(-6\mathbf{i}+5\mathbf{j}\)

2. (b) \(\begin{pmatrix}4\\-2\\0\end{pmatrix}\)

   • The three numbers go with i, j and k
   • A zero component gives no term
   • \(4\mathbf{i}-2\mathbf{j}\)

Write each vector as a column vector with three components.

1. (a) \(2\mathbf{i}-5\mathbf{j}+\mathbf{k}\)

   • Read the coefficients of i, j and k in that order
   • Every term is there
   • \(\begin{pmatrix}2\\-5\\1\end{pmatrix}\)

2. (b) \(4\mathbf{j}-9\mathbf{k}\)

   • Read the coefficients of i, j and k in that order
   • There is no i term, so that component is 0
   • \(\begin{pmatrix}0\\4\\-9\end{pmatrix}\)

3. (c) \(-5\mathbf{i}+6\mathbf{k}\)

   • Read the coefficients of i, j and k in that order
   • There is no j term, so that component is 0
   • \(\begin{pmatrix}-5\\0\\6\end{pmatrix}\)

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.

   • East and north are positive, west and south are negative
   • First component: east/west. Second component: north/south
   • \(\begin{pmatrix}-9\\4\end{pmatrix}\)

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

   • Coefficient of i is the first component, coefficient of j the second
   • \(-9\mathbf{i}+4\mathbf{j}\)

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.

   • East, north and up are positive. West, south and down are negative
   • \(\begin{pmatrix}40\\-25\\-12\end{pmatrix}\)

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

   • The three components are the coefficients of i, j and k
   • \(40\mathbf{i}-25\mathbf{j}-12\mathbf{k}\)

3. (c) The submarine's velocity is 3 m/s towards the north. Is velocity a vector or a scalar?

   • Velocity has a size and a direction, so it is a vector

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 \(q\).

• Write the second vector as a column: \(\begin{pmatrix}-11\\6\\4\end{pmatrix}\)
• Equal vectors have equal components: \(3p-2=-11\) and \(3-q=4\)
• \(3p=-9,\ p=-3\) and \(q=-1\)

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

• Count from the tail to the head: right 5, then down 3
• \(\mathbf{v}=\begin{pmatrix}5\\-3\end{pmatrix}\)
• The components are the coefficients of i and j: \(\mathbf{v}=5\mathbf{i}-3\mathbf{j}\)