Position vectors, displacement vectors and distance Problem set A

Points \(A(3, -5, 2)\) and \(B(-1, 4, 7)\) are given. Find:

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

Points A and B have position vectors \(\mathbf{a}=2\mathbf{i}+7\mathbf{j}-3\mathbf{k}\) and \(\mathbf{b}=5\mathbf{i}-\mathbf{j}+4\mathbf{k}\). 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) \(\mathbf{a}=3\mathbf{i}-2\mathbf{j}+\mathbf{k},\ \mathbf{b}=\mathbf{i}+4\mathbf{j}-5\mathbf{k};\)
2. (b) \(\mathbf{a}=2\mathbf{i}+\mathbf{j},\ \mathbf{b}=5\mathbf{i}-3\mathbf{j}+\mathbf{k}.\)

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

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

Relative to a control tower at O, drone P has position vector \(\mathbf{p}=\begin{pmatrix}40 \\ -25 \\ 60\end{pmatrix}\) and drone Q has position vector \(\mathbf{q}=\begin{pmatrix}-10 \\ 30 \\ 20\end{pmatrix}\), in metres, where \(\mathbf{k}\) is vertically up. Find the distance \(PQ\) between the drones to 3 significant figures.

A hiker starts at the point \(A(2,-3,5)\) and walks to a point B. The displacement is \(\overrightarrow{AB}=\begin{pmatrix}4 \\ 1 \\ -6\end{pmatrix}\). Find the position vector of B.

The points \(A(3, -2, 4)\), \(B(-3, -3, 2)\) and \(C(2, 4, 6)\) 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 \(A(3, 6, 2)\), \(B(-3, 5, 0)\) and \(C(-1, -3, -2)\) 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.