Position vectors, displacement vectors and distance Problem set B

Points \(A(-4, 2, 6)\) and \(B(5, -3, 1)\) are given. Find:

1. (a) \(\overrightarrow{AB}\);
2. (b) \(\overrightarrow{BA}\).

Points A and B have position vectors \(\mathbf{a}=-3\mathbf{i}+4\mathbf{j}+\mathbf{k}\) and \(\mathbf{b}=\mathbf{i}+\mathbf{j}-6\mathbf{k}\). Write \(\overrightarrow{AB}\) in the form \(x\mathbf{i}+y\mathbf{j}+z\mathbf{k}\).

Find the exact distance \(AB\) between points A and B with position vectors:

1. (a) \(\mathbf{a}=4\mathbf{i}-\mathbf{j}+2\mathbf{k},\ \mathbf{b}=5\mathbf{j}-2\mathbf{k};\)
2. (b) \(\mathbf{a}=\mathbf{i}+3\mathbf{j},\ \mathbf{b}=3\mathbf{i}-\mathbf{j}+4\mathbf{k}.\)

Find the exact distance \(AB\) between points A and B with position vectors:

1. (a) \(\mathbf{a}=\begin{pmatrix}1 \\ 4 \\ -2\end{pmatrix},\ \mathbf{b}=\begin{pmatrix}7 \\ 2 \\ 1\end{pmatrix};\)
2. (b) \(\mathbf{a}=\begin{pmatrix}0 \\ -3 \\ 5\end{pmatrix},\ \mathbf{b}=\begin{pmatrix}2 \\ -6 \\ 6\end{pmatrix}.\)

Relative to the corner O of a fish tank, a goldfish P has position vector \(\mathbf{p}=\begin{pmatrix}18 \\ 22 \\ 7\end{pmatrix}\) and a snail Q has position vector \(\mathbf{q}=\begin{pmatrix}-6 \\ 10 \\ 35\end{pmatrix}\), in centimetres, where \(\mathbf{k}\) is vertically up. Find the distance \(PQ\) between them to 3 significant figures.

A skier starts at the point \(A(-5,8,2)\) and ends at a point B. The displacement is \(\overrightarrow{AB}=\begin{pmatrix}3 \\ -4 \\ 9\end{pmatrix}\). Find the position vector of B.

The points \(A(3, 1, 1)\), \(B(0, 4, 5)\) and \(C(-2, -2, 1)\) are the corners of a triangle.

1. (a) Find the exact lengths \(AB\), \(AC\) and \(BC\).
2. (b) Show that the triangle is isosceles.

The points \(A(2, 3, 2)\), \(B(-2, 4, 0)\) and \(C(0, 4, -4)\) are the corners of a triangle.

1. (a) Find \(AB^2\), \(BC^2\) and \(AC^2\).
2. (b) Show that angle \(ABC\) is a right angle.