Geometry: vector equation of a plane

[Kate2010]

Homework Statement

A plane contains non-collinear points A,B,C with position vectors a,b,c with respect to O. If s is the position vector of a point P, in the plane, with respect to O, show that it may be written as s = ua + vb + wc where u+v+w=1 (I think I have done this). If OP is perpendicular to the plane show

u = n.(bxc)/|n|²

v= n.(cxa)/|n|²

w= n.(axb)/|n|²

where n= bxc + cxa + axb, . is for the scalar product, x is for the vector product

Homework Equations

The Attempt at a Solution

I have shown that n is normal to the plane. In an earlier part of the question I was asked to find u,v and w in terms of s,a,b,c. I found, for example, that u= [(b-c)x(s-c)]/[(b-c)x(a-c)], working from s= c + u(a-c) + v(b-c). I'm not even sure if this part is correct. Any advice on what to do would be great.

 

[mighty2000]

I have reviewed your work and it seems like you have made some progress in solving the problem. To find u, v, and w in terms of s, a, b, and c, we can start by setting up the equation s = ua + vb + wc and solving for u, v, and w.

To find u, we can take the dot product of both sides of the equation with vector (a-c). This gives us:

(a-c)•s = (a-c)•(ua + vb + wc)

Expanding the dot product on the left side, we get:

(a•s)-(c•s) = u(a•a) + v(a•b) + w(a•c) - u(c•a) - v(c•b) - w(c•c)

Since we know that u + v + w = 1, we can substitute this into the equation to get:

(a•s)-(c•s) = (a•a) - (c•c) + (a•b - c•b)v + (a•c - c•a)w

Similarly, we can find expressions for v and w by taking the dot product with vectors (b-c) and (c-a) respectively. This gives us:

(b•s)-(c•s) = (b•b) - (c•c) + (b•a - c•a)u + (b•c - c•b)w

(c•s)-(a•s) = (c•c) - (a•a) + (c•b - a•b)u + (c•a - a•c)v

We now have a system of three equations with three unknowns (u, v, and w). We can solve for these unknowns using any appropriate method (substitution, elimination, etc.). Once we have found the values for u, v, and w, we can substitute them back into the original equation s = ua + vb + wc to get the desired expression.

I hope this helps. Keep up the good work with your mathematical reasoning and problem-solving skills!