# Maths vectors question

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Thread starter 4 years ago
#1
The flight path of a helicopter H taking off from an airport.
Coordinate axes Oxyz are set up with the origin O at the base of the airport control tower. The x-
axis is due east, the y-axis due north, and the z-axis vertical. The units of distance are kilometres
throughout.
The helicopter takes off from the point G. The position vector r of the helicopter t minutes after
take-off is given by: r=(1+t)i+(0.5+2t)j+2tk .

(i) Write down the coordinates of G - (1,0.5.0)
(ii) Find the angle the flight path makes with the horizontal.

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4 years ago
#2
(Original post by avsl020)
The flight path of a helicopter H taking off from an airport.
Coordinate axes Oxyz are set up with the origin O at the base of the airport control tower. The x-
axis is due east, the y-axis due north, and the z-axis vertical. The units of distance are kilometres
throughout.
The helicopter takes off from the point G. The position vector r of the helicopter t minutes after
take-off is given by: r=(1+t)i+(0.5+2t)j+2tk .

(i) Write down the coordinates of G - (1,0.5.0)
(ii) Find the angle the flight path makes with the horizontal.
What have you tried so far? A diagram may help, and to draw a triangle on that diagram.
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Thread starter 4 years ago
#3
I tried the cosine rule but I didnt get very far because the vector is in terms of t, and they dont cancel out

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4 years ago
#4
(Original post by avsl020)
I tried the cosine rule but I didnt get very far because the vector is in terms of t, and they dont cancel out

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It's quite simple if you just take the position of H to be at an arbitrary time such as , and then you can deduce the coordinates for F.

Then EDIT: You can also do it all in terms of but it must obviously cancel out.
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4 years ago
#5
(Original post by avsl020)
I tried the cosine rule but I didnt get very far because the vector is in terms of t, and they dont cancel out
Well, can you find a vector that is in the same direction as the flight path? (You should actually be able to write one down without any work)!

Call that vector v and split out the horizontal and vertical components. (i.e. the part in the xy plane and the part that's vertical).

What's the size of the horizontal component?
What's the size of the vertical component?

So the angle the vector makes with the horizontal is going to be?
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Thread starter 4 years ago
#6
So coordinates of F: (0, 2.5, 0) ?

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4 years ago
#7
(Original post by avsl020)
So coordinates of F: (0, 2.5, 0) ?

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The coordinates of F would be the same as H except the z-coordinate is 0.
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Thread starter 4 years ago
#8
Ok, so when t=1, the vector G to H is 2i+2.5j+2k,
Coordinates of H: (2, 2.5, 2)
Coordinates of F: (2, 2.5, 0)

scalar product= (2×2)+(2.5*2.5)= 10.25

|GH|= square root of (4+6.25+4) = root 54/2

|GF|= square root of (4+6.25) = root 10.23

When i sub everything into the equation and then do the inverse to work out theta, i get 31.99 degrees which is the wrong answer (it's 41.8)
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4 years ago
#9
(Original post by avsl020)
Ok, so when t=1, the vector G to H is 2i+2.5j+2k,
Coordinates of H: (2, 2.5, 2)
Coordinates of F: (2, 2.5, 0)

scalar product= (2×2)+(2.5*2.5)= 10.25

|GH|= square root of (4+6.25+4) = root 54/2

|GF|= square root of (4+6.25) = root 10.23

When i sub everything into the equation and then do the inverse to work out theta, i get 31.99 degrees which is the wrong answer (it's 41.8)
You don't really need to carry out any scalar products.

You simply need to find and by Pythagoras' Theorem, then by trigonometry (and aid of the diagram) you know that Just to add on, your and are incorrect
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10 months ago
#10
avsl020Coordinate G(1,0.5,0)Vector GH = i 2j 2kSo coordinate F(2,2.5,0)Vector GF =i 2jScalar Product GH x GF = 5Magnitude GF =square root 5Magnitude GH =3So the angle of the flight path is 41.8
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10 months ago
#11
Coordinate G(1,0.5,0)
Vector GH = i+2j+2k
So coordinate F(2,2.5,0)
Vector GF =i+2j
Magnitude GF =square root 5
Magnitude GH=3
Scalar Product GF x GH=5
Cos x=Scalar Product/Magnitude GH x Magnitude GF
So x,the angle of the flight is 41.8 to 1 decimal place
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