Vector product

The points P, Q, R and S have coordinates (1, 2, 4), (3, 1, 6), (4, 8, - 2) and (5, 5, 5) respectively.

Find, in exact form, the area of triangle PQR.

Find, in exact form, the volume of tetrahedron PQRS.

Use your answers to part (a) and part (b) to deduce the shortest distance from S to the plane containing P, Q and R. Give your answer in exact form.

[You may use the formula V = Ah, where V is the volume of a pyramid, A is the area of one

of its bases and h is the height relative to the base.]

The points P, Q and R have coordinates (0, 2, 3), (2, 0, 1) and (1, 3, 0) respectively.

The acute angle between the line segments PQ and PR is θ.

The triangle PQR lies in the plane Π.

Determine an equation for Π, giving your answer in the form ax + by + cz = d, where a, b, c

and d are integers.

The points A and B have position vectors a = i + p j + q k and b = 2 i + 3 j + 2 k respectively, relative to the origin O.

Determine the value of p and the value of q for which a × b = 2 i + 6 j - 11 k.

The point C has coordinates (d, e, f) and the tetrahedron OABC has volume 7.

Using the values of p and q found in part (a), find the possible relationships between

d, e and f.

Explain the geometrical significance of these relationships.

You are given the variable point A(3, 8, t), where t is a real parameter, and the fixed point B(1, 2, 2).

Use the vector product to find an expression, in terms of t, for the area of triangle OAB.

Hence determine the value of t for which the area of triangle OAB is a minimum.

Find the position vector of the point of intersection of l₁ and Π.

Find the acute angle between l₁ and Π.

A is the point on l₁ where λ = 1.

l₂ is the line with the following properties.

• l₂ passes through A

• l₂ is perpendicular to l₁

• l₂ is parallel to Π

Find, in vector form, the equation of l₂.

The points P, Q and R have position vectors p = 2 i + j + 5 k, q = i - j + k and r = 2 i + j + t k respectively, relative to the origin O.

Determine the value(s) of t in each of the following cases.

The line OR is parallel to p × q.

The volume of tetrahedron OPQR is 13.