Find the Equation of Plane & Distance from Point

[VooDoo]

Hi Guys,

I am stuck on a maths question for my first year of Engineering at University. Your help would be greatly appreciated!

I don't know where to start

1. Given that a = (3, 1, 2) and b = (1,−2,−4) are the position vectors of the points P and Q
respectively, find
(a) the equation of the plane passing through Q and perpendicular to PQ,
(b) the distance rom the point (−1, 1, 1) to the plane obtained in (a).

Thanks guys

 

[HallsofIvy]

You need to know: The equation of a plane containing $(x_0,y_0,z_0)$, perpendicular to the vector Ai+ Bj+ Ck, is $A(x-x_0)+ B(y- y_0)+ C(z- z_0)= 0$. (For any point (x,y,z) in that plane, $(x-x_0)i+ (y- y_0)j+ (z- z_0)k$ is in the plane and so is perpendicular to $Ai+ Bj+ Ck$. The dot product of the two is 0.)

In this problem, the vector PQ is (3-1)i+ (1-(-2))j+ (2-(-4))k= 2i+ 3j+ 6k

As far as (b) is concerned, I'll bet there is a formula for the distance between a point and a plane in this same section of your textbook. To do it without that formula, remember that the shortest distance between a point and a plane is along a line perpendicular to that plane.
i) Write the parametric equations, for x, y, z in terms of a paratmeter t, for a line through (-1, 1, 1) in the same direction as the vector 2i+ 3j+ 6k.
ii) Plug those equations into the equation of the plane, from (a), to get a single equation in the single variable t and solve for t.

iii) Put that value of t into the parametric equations to get (x,y,z) coordinates of the point on the line and plane.

iii) Calculate the distance between that point and (-1, 1, 1).