Vector product

The parallelogram P has vertices E, F, G and H, with coordinates

Determine the exact area of P, giving your answer as a simplified surd.

The line l passes through E and is perpendicular to the plane containing P

Determine an equation for l giving your answer in the form (r – a)×b = 0

where a and b are constant vectors.

Given that

• P is one face of a parallelepiped

• the opposite face of the parallelepiped lies in the plane Π

• the point (25, –4, 13) lies in Π

determine the volume of the parallelepiped.

The line l₁ has equationr 3 λ 4

where λ and μ are scalar parameters.

The lines l₁ and l₂ intersect at the point P.

Determine the coordinates of P.

Given that the plane Π contains both l₁ and l₂

determine a Cartesian equation for Π.

Determine a Cartesian equation for each of the two lines that

• pass through (0, 0, 0)

• make an angle of 60° with the x-axis

• make an angle of 45° with the y-axis

With respect to a fixed origin O the point A has coordinates (3, 6, 5) and the line l has equation

The points B and C lie on l such that AB = AC = 15

Given that A does not lie on l and that the x coordinate of B is negative,

determine the coordinates of B and the coordinates of C

Hence determine a Cartesian equation of the plane containing the points A, B and C

The point D has coordinates (–2, 1, α), where α is a constant.

Given that the volume of the tetrahedron ABCD is 147

determine the possible values of α

Given that α 0

determine the shortest distance between the line l and the line passing through the points A and D, giving your answer to 2 significant figures.

With respect to a fixed origin O, the points A and B have coordinates (2, 2, –1) and (4, 2 p, 1) respectively, where p is a constant.

For each of the following, determine the possible values of p for which,

OB makes an angle of 45° with the positive x-axis

A small aircraft is landing in a field.

In a model for the landing the aircraft travels in different straight lines before and after it lands, as shown in Figure 2.

The vector v_A is in the direction of travel of the aircraft as it approaches the field. The vector v_L is in the direction of travel of the aircraft after it lands. With respect to a fixed origin, the field is modelled as the plane with equation

x – 2 y + 25 z = 0

and

Write down a vector n that is a normal vector to the field.

When the aircraft lands it remains in contact with the field and travels in the direction v_L The vector v_L is in the same plane as both v_A and n as shown in Figure 2.

Determine a vector which has the same direction as v_L

State a limitation of the model.

The points A, B and C, with position vectors a= 3i− 2j+k, b=i+ 4j+ 5k and c = −2 i + 3 j + 3 k respectively, lie on the plane Π

Find ABAC

Find an equation for Π in the form r.n=p

The point D has position vector 8i+ 7j+ 5k

Determine the volume of the tetrahedron ABCD

With respect to a fixed origin O, the points A, B and C have coordinates (3, 4, 5), (10, –1, 5) and (4, 7, –9) respectively.

The plane has equation 4x – 8y + z = 2

The line segment AB meets the plane at the point P and the line segment BC meets the plane at the point Q.

Show that, to 3 significant figures, the area of quadrilateral APQC is 38.5

The point D has coordinates (k, 4, –1), where k is a constant. Given that the vectors AB, AC and AD form three edges of a parallelepiped of volume 226

find the possible values of the constant k.