Parallel vectors, perpendicular vectors and area Problem set B

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

Find the area of each shape.

1. (a) The triangle with vertices at the points with position vectors \(\mathbf{a}=2\mathbf{i} + 2\mathbf{j} + \mathbf{k}\), \(\mathbf{b}=3\mathbf{j} + 4\mathbf{k}\) and \(\mathbf{c}=5\mathbf{i} + \mathbf{j} + 3\mathbf{k}\). Find its area.
2. (b) The parallelogram with sides \(\mathbf{u}=3\mathbf{i} + \mathbf{j} - 2\mathbf{k}\) and \(\mathbf{w}=-\mathbf{i} + 2\mathbf{j} + 4\mathbf{k}\). Find its area.

The points \(P(0, 4, 1)\), \(Q(2, 5, 3)\), \(R(6, 7, 7)\) and \(S(1, 1, 3)\) 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 sail for a dinghy is a triangle with corners \(P(2, 0, 0)\), \(Q(5, 3, 1)\), \(R(1, 4, 2)\), in metres. The sailcloth costs \$23 per square metre.

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

A triangular field on a hillside has its corners at \(A(0, 2, 1)\), \(B(4, 5, 3)\), \(C(1, 7, 6)\), in metres. A fence is to be built along the edge AB.

1. (a) Find the area of triangle ABC.
2. (b) Find the shortest distance from the corner C to the fence (the line AB).

The triangle has vertices \(A(2, 0, 0)\), \(B(0, 3, 0)\) and \(C(0, 0, k)\), where \(k>0\). The area of the triangle is 6 square units.

1. (a) Show that \(|\overrightarrow{AB}\times\overrightarrow{AC}|^2=13 k^{2} + 36\).
2. (b) Find the value of \(k\).