Equation of a plane

Verify that a cartesian equation of Π₁ is x - y + 2z = 3.

For some real constant a, cartesian equations of planes Π₂ and Π₃ are

Π₂: x - 3z = 1 Π₃: ax - y - z = 4

By considering a suitable matrix, show that Π₁, Π₂ and Π₃ intersect at a single point for all

values of a except a = 2.

Use the matrix from part (b) to find the coordinates of the point of intersection of Π₁, Π₂

and Π₃ in the case where a = 3.

In the case where a = 2, determine the geometrical arrangement of Π₁, Π₂ and Π₃.

In this question you must show detailed reasoning.

The surface S is given by the equation z = 4x²y - 3xy² - 5x for all real values of x and y.

The point P (1, 1, - 4) lies on S.

Find the equation of the tangent plane to S at P.

The point Q is a stationary point of S with positive x - and y-coordinates.

Find, in exact form, the coordinates of Q.

Find the value of the determinant of H, the Hessian matrix of S, at Q.

Deduce the nature of the stationary point Q.

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.

Find the equation of the tangent plane to E at P, giving your answer in the form

ax + by + cz = d where a, b, c and d are integers.

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.