[the_anomaly]

Im stuck on question 61 part (c) of the P6 Heinemann Book Review Exercise. If anyone can show me how to complete I would be very grateful. For anyone that doesnt have the book the question is:

The plane II passes through A(3, -5, -1), B(-1, 5, 7) and C(2, -3, 0).

a) Find AC x BC

b) Hence, or otherwise, find an equation, in the form r.n = p, of the plane II.

c) The perpendicular from the point (2, 3, -2) to II meets the plane at P. Find the co-ordinates of P.

Thanks in advance.

[Gaz031]

Originally Posted by the_anomaly

Im stuck on question 61 part (c) of the P6 Heinemann Book Review Exercise. If anyone can show me how to complete I would be very grateful. For anyone that doesnt have the book the question is:

The plane II passes through A(3, -5, -1), B(-1, 5, 7) and C(2, -3, 0).

a) Find AC x BC

b) Hence, or otherwise, find an equation, in the form r.n = p, of the plane II.

c) The perpendicular from the point (2, 3, -2) to II meets the plane at P. Find the co-ordinates of P.

Thanks in advance.

'The perpendicular' is a line passing through (2,3,-2) which is perpendicular and hence parallel to 'n' (which you found earlier).
Thus, you can write the equation of the line in the form r=a+[Mimetex cannot convert this formula]b.
You can then rewrite the equation of the line in the form r=i(..)+j(..)+k(..).
So you can then say a typical point on the line has coordinates (.. , .. , ..).
You can then take this typical point to be 'r' and put it in the equation of the plane and do the dot product.
Solve this equation to find your variable [Mimetex cannot convert this formula] at P, where the line and the plane intersect.
Put this value of [Mimetex cannot convert this formula] back into your typical point coordinates and hence find the coordinates of P.

[Fermat]

let n be the normal to the plane
D = (2,3,-2)
then
r = D + λn
is the eqn of the perpindicular, to the palne, through D

find out where the line r cuts the plane.