Equation of a plane

Find a vector that is normal to the plane ∏ 5 (a)

Find a Cartesian equation of the plane ∏ 5 (b)

The position vectors of the points A, B and C are

respectively.

Find the area of the triangle ABC

The points A, B and C all lie in the plane

Find an equation of the plane , in the form r n d

The point P has position vector p i 4j 2k

Find the exact distance of P from

In this question all measurements are in centimetres.

A small, thin laser pen is set up with one end at A(7, 2, 3) and the other end at

A laser beam travels from A to B and continues in a straight line towards a large thin sheet of glass.

The sheet of glass lies within a plane ₁ which is modelled by the equation

where p is an integer.

A second large sheet of glass lies on the other side of ₁

This second sheet lies within a plane ₂ which is modelled by the equation

Calculate the distance between the sheets of glass.

The point A(7, 2, 3) is reflected in ₁

Find the coordinates of the image of A after reflection in ₁

Find the acute angle between the lines L₁ and L₂, giving your answer to the

Find the equation of ₁, giving your answer in the form r.n d

Hence find the shortest distance of the plane ₁ from the origin.

Three planes have equations

The planes do not meet at a unique point.

Show that k 45 is one possible value of k, and find the other possible value of k.

For each value of k found in part (a), identify the configuration of the given planes.

In each case fully justify your answer, stating whether or not the equations of the planes form a consistent system.

Three planes have equations

where k is a real constant.

The planes do not meet at a unique point.

Find the possible values of k.

For each value of k found in part (a), identify the configuration of the given planes.

Fully justify your answer, stating in each case whether or not the equations of the planes form a consistent system.

The points A, B and C have coordinates A(4, 5, 2), B(3, 2, 4) and C(2, 6, 1)

The points A, B and C lie in a plane.

Find a vector equation of the plane in the form r.n k

Hence find the exact distance of the plane from the origin.