Scalar product

Show that the angle between L₁ and L₂ is 0.701 radians, correct to three decimal places.

Find the shortest distance between L₁ and L₂

Give your answer in an exact form.

It is given that

where λ is a constant.

Find the value of λ

Circle your answer.

–28 –8 8 28

The line l₁ passes through the points A(6, 2, 7) and B(4, 3, 7)

Find a Cartesian equation of l₁

Explain how you know that the lines l₁ and l₂ are not perpendicular.

The lines l₁ and l₂ both lie in the same plane.

Find the value of c

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.

State any restrictions on the value of p.

The equations below represent three planes.

Find, in terms of p, the coordinates of the point of intersection of the three planes.

In the case where p 2, show that the planes are mutually perpendicular.

Which of the following lines is perpendicular to the line L ?

Tick () one box.

Calculate the acute angle between the line and the plane.

Give your answer to the nearest 0.1

The points A(5, 4, 6) and B(6, 6, 8) lie on the line L. The point C is (15, 5, 9).

D is the point on L that is closest to C.

Find the coordinates of D.

Hence find, in exact form, the shortest distance from C to L.