Equation of a plane

An archer shoots an arrow towards a target.

In a model

• the arrow is a particle

• the flight path of the arrow is a straight line

• the target is part of a plane

Relative to a fixed origin O

• the arrow is fired from the point with position vector 3i 5j 2k

• the plane containing the target has equation 2x 4y z 3

Use the model to answer parts (a) to (d).

Determine the shortest distance that the arrow must travel to reach the plane.

The arrow hits the target at the point with position vector 6i 2j k

Determine a vector equation of the flight path of the arrow.

Determine the acute angle that the flight path of the arrow makes with the target. Give your answer to the nearest degree.

Determine the distance travelled by the arrow.

Comment on whether the actual distance travelled by the arrow is likely to match the answer to part (d), giving a reason for your answer.

The line l₁ has equation

r = i – 2 j + 3 k + λ (2 i + j – 4 k)

and the line l₂ has equation

r = 5 i + p j – 7 k + μ (6 i + j + 8 k)

where λ and μ are scalar parameters and p is a constant.

The plane Π contains l₁ and l₂

Show that the vector 3i – 10j – kis perpendicular to Π

Hence determine a Cartesian equation of Π

Hence determine the value of p

Given that

• the lines l₁ and l₂ intersect at the point A

• the point B has coordinates (12, –11, 6)

determine, to the nearest degree, the acute angle between AB and Π

The line l₁ has equation

r = i – 2 j + 3 k + λ (2 i + j – 4 k)

and the line l₂ has equation

r = 5 i + p j – 7 k + μ (6 i + j + 8 k)

where λ and μ are scalar parameters and p is a constant.

The plane Π contains l₁ and l₂

Show that the vector 3i – 10j – kis perpendicular to Π

Hence determine a Cartesian equation of Π

Hence determine the value of p

Given that

• the lines l₁ and l₂ intersect at the point A

• the point B has coordinates (12, –11, 6)

determine, to the nearest degree, the acute angle between AB and Π

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.

Two birds are flying towards their nest, which is in a tree.

Relative to a fixed origin, the flight path of each bird is modelled by a straight line.

In the model, the equation for the flight path of the first bird is

and the equation for the flight path of the second bird is

where λ and μ are scalar parameters and a is a constant.

In the model, the angle between the birds’ flight paths is 120°

Determine the value of a.

Verify that, according to the model, there is a common point on the flight paths of the two birds and find the coordinates of this common point.

The position of the nest is modelled as being at this common point.

The tree containing the nest is in a park.

The ground level of the park is modelled by the plane with equation

2 x – 3 y + z = 2

Hence determine the shortest distance from the nest to the ground level of the park.

By considering the model, comment on whether your answer to part (c) is reliable, giving a reason for your answer.

Two birds are flying towards their nest, which is in a tree.

Relative to a fixed origin, the flight path of each bird is modelled by a straight line.

In the model, the equation for the flight path of the first bird is

and the equation for the flight path of the second bird is

where λ and μ are scalar parameters and a is a constant.

In the model, the angle between the birds’ flight paths is 120°

Determine the value of a.

Verify that, according to the model, there is a common point on the flight paths of the two birds and find the coordinates of this common point.

The position of the nest is modelled as being at this common point.

The tree containing the nest is in a park.

The ground level of the park is modelled by the plane with equation

2 x – 3 y + z = 2

Hence determine the shortest distance from the nest to the ground level of the park.

By considering the model, comment on whether your answer to part (c) is reliable, giving a reason for your answer.

The points P, Q and R have position vectors 2, 1 and 0 respectively.

Determine a vector equation of the plane that passes through the points P, Q and R, giving your answer in the form r = a + λ b + μ c, where λ and μ are scalar parameters.

Determine the coordinates of the point of intersection of the plane with the x-axis.

The plane Π has equation

where λ and μ are scalar parameters.

Show that vector 2i + 3j – 4k is perpendicular to Π.

Hence find a Cartesian equation of Π.

The line l has equation

where t is a scalar parameter.

The point A lies on l.

determine the possible coordinates of A.

The plane Π has equation

3	1	2
3	2	0

where λ and μ are scalar parameters.

Show that vector 2i + 3j – 4k is perpendicular to Π.

Hence find a Cartesian equation of Π.

The line l has equation

where t is a scalar parameter.

The point A lies on l.

determine the possible coordinates of A.

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 plane Π₁ has equation

r = 2 i + 4 j – k + λ (i + 2 j – 3 k) + μ (– i + 2 j + k)

where λ and μ are scalar parameters.

Find a Cartesian equation for Π₁

The line l has equation

Find the coordinates of the point of intersection of l with Π₁

The plane Π₂ has equation

r. (2 i – j + 3 k) = 5

Find, to the nearest degree, the acute angle between Π₁ and Π₂

The plane Π₁ has equation

r = 2 i + 4 j – k + λ (i + 2 j – 3 k) + μ (– i + 2 j + k)

where λ and μ are scalar parameters.

Find a Cartesian equation for Π₁

The line l has equation

Find the coordinates of the point of intersection of l with Π₁

The plane Π₂ has equation

r. (2 i – j + 3 k) = 5

Find, to the nearest degree, the acute angle between Π₁ and Π₂

Given that k ≠4, find, in terms of k, the inverse of the matrix M.

Find, in terms of p, the coordinates of the point where the following planes intersect.

2 x + 5 y + 7 z = 1 x + y + z = p 2 x + y − z = 2

Find the value of q for which the following planes intersect in a straight line.

4 x + 5 y + 7 z = 1 x + y + z = q 2 x + y − z = 2

For this value of q, determine a vector equation for the line of intersection.

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

The line l has equation

The line l₂ has equation

where t is a scalar parameter.

Show that l and l₂ lie in the same plane.

Write down a vector equation for the plane containing l and l₂

Find, to the nearest degree, the acute angle between l and l₂

The line l has equation

The line l₂ has equation

where t is a scalar parameter.

=Show that l and l₂ lie in the same plane.

Write down a vector equation for the plane containing l and l₂

=Find, to the nearest degree, the acute angle between l and l₂

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.