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Further Maths Vectors year 1 question

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
r1 = (-i + 5j + 2k) + λ(2i + aj)
and the equation for the flight path of the second bird is
r2 = (4i - j + 3k) + μ( j - k)
where λ and μ are scalar parameters and a is a constant.
In the model, the angle between the birds’ flight paths is 120°
(a) Determine the value of a.
(b) 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
2x 3y + z = 2
(c) Hence determine the shortest distance from the nest to the ground level of the park.
(d) By considering the model, comment on whether your answer to part (c) is reliable,
giving a reason for your answer.


This is the question and I get the answer for a, however when a^2 is 4, it only accepts -2, can anyone explain why?
Reply 1
Original post by SGDumpling
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
r1 = (-i + 5j + 2k) + λ(2i + aj)
and the equation for the flight path of the second bird is
r2 = (4i - j + 3k) + μ( j - k)
where λ and μ are scalar parameters and a is a constant.
In the model, the angle between the birds’ flight paths is 120°
(a) Determine the value of a.
(b) 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
2x 3y + z = 2
(c) Hence determine the shortest distance from the nest to the ground level of the park.
(d) By considering the model, comment on whether your answer to part (c) is reliable,
giving a reason for your answer.
This is the question and I get the answer for a, however when a^2 is 4, it only accepts -2, can anyone explain why?
When a=2, the angle is 60, not 120. It would help to see what you did for the extraneous solution a=2.
(edited 2 months ago)
Reply 2
Original post by mqb2766
When a=2, the angle is 60, not 120. It would help to see what you did if you for the extraneous solution a=2.
oh my god thank you so much! i didnt think of that but if i did plug 2 into the equation, cosθ is 1/2 which is 60 degrees. Thank you so much
Reply 3
Original post by SGDumpling
oh my god thank you so much! i didnt think of that but if i did plug 2 into the equation, cosθ is 1/2 which is 60 degrees. Thank you so much
Exactly. The r1.r2=a must be negative for an obtuse angle.
(edited 2 months ago)

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