Vector equation of a line

Show that the cables do not intersect.

To access the cables for maintenance, a ladder can be used. The base of the ladder is placed at a fixed point on the ground.

The ladder is modelled as a straight-line segment. The base of the ladder is modelled as being located at the point (4, 5, 0).

Determine the minimum length of the ladder required so that it reaches cable T₁. Give your

answer in centimetres to the nearest centimetre.

Identify a modelling assumption used in part (a) that is unrealistic, and which could affect

your answer to this part.

Find, in the same form, a vector equation of the line segment that represents the wire T. The components of the direction vector should be integers whose only positive common factor is 1.

For the construction site to be considered safe, it must pass two tests.

Test 1: The wires S and T need to be at least 5 metres apart at all positions on S and T.

By using an appropriate formula, determine whether the construction site passes Test 1.

A security camera is placed at a point Q on wire S.

Test 2: To ensure sufficient visibility of the construction site, the distance between the security camera and the top of P₃ must be at least 19 m.

Determine whether it is possible to find point Q on S such that the construction site passes

Test 2.

Points A, B and C have coordinates (4, 2, 0), (1, 5, 3) and (1, 4, - 2) respectively. The line l passes through A and B.

Find a cartesian equation for l.

M is the point on l that is closest to C.

Find the coordinates of M.

Find the exact area of the triangle ABC.

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.