Parallel vectors, perpendicular vectors and area Problem set A

Find the area of the triangle with vertices \(A(1, 2, 3)\), \(B(3, 0, 4)\), \(C(2, 5, 1)\).

Find the area of each shape.

1. (a) The triangle with vertices at the points with position vectors \(\mathbf{a}=\mathbf{i} + 2\mathbf{k}\), \(\mathbf{b}=3\mathbf{i} + \mathbf{j}\) and \(\mathbf{c}=2\mathbf{i} + 4\mathbf{j} + 3\mathbf{k}\). Find its area.
2. (b) The parallelogram with sides \(\mathbf{u}=2\mathbf{i} - \mathbf{j} + 3\mathbf{k}\) and \(\mathbf{w}=\mathbf{i} + 4\mathbf{j}\). Find its area.

The points \(P(1, 2, 0)\), \(Q(3, 3, 2)\), \(R(7, 5, 6)\) and \(S(2, 6, 1)\) are given.

1. (a) Show that P, Q and R lie on a straight line, by finding \(\overrightarrow{PQ}\times\overrightarrow{PR}\).
2. (b) Find the area of the triangle PQS.

A solar roof panel is a triangle with corners \(P(0, 0, 2)\), \(Q(3, 1, 5)\), \(R(1, 4, 3)\), in metres. The panel costs \$64 per square metre.

1. (a) Find the area of the roof panel.
2. (b) Find the cost.

A triangular flower bed has its corners at \(A(1, 1, 0)\), \(B(6, 2, 1)\), \(C(3, 6, 4)\), in metres. A path is to run along the edge AB.

1. (a) Find the area of triangle ABC.
2. (b) A gardener stands at C. Find the shortest distance from C to the path (the line AB).

The triangle has vertices \(A(1, 0, 0)\), \(B(0, 2, 0)\) and \(C(0, 0, k)\), where \(k>0\). The area of the triangle is 5 square units.

1. (a) Show that \(|\overrightarrow{AB}\times\overrightarrow{AC}|^2=5 k^{2} + 4\).
2. (b) Find the value of \(k\).