The question: Consider the triangle AOPQ as shown below. Let p = Op rightarrow a

Question

The question:

Consider the triangle AOPQ as shown below. Let p = Op rightarrow and q = OQ rightarrow . Write down an expression in terms of p and q for a vector which points directly out of the page and has magnitude equal to the area of delta OPQ. (Hint: consider a cross product, and be careful about using the right hand rule.) Consider the tetrahedron OPQR as shown below. Let p = Op rightarrow, q = OQ rightarrow , and r = OR rightarrow . Write down expressions for the vectors PQ rightarrow , PR rightarrow , QR rightarrow in terms of p, q, r. Let a be a vector with length equal to the area of delta OPQ which points out perpendicularly from the face OPQ. Let vectors b, c, d be defined analogously with respect to the faces OPR, ORQ, PQR (in that order). Write down expressions for a. b, c, d in terms of p. q. r. Then show that a + b + c + d = 0. Now suppose that POQ = POR = ROQ = 90 degree . Let A,B,C,D denote the areas of the faces OPQ. OPR, ORQ, PQR (in that order). Show that A2 + B2 + C2 = D2. This is another kind of 3D version of Pythagoras' Theorem. (Hint: One way is to consider the dot product of a + b + c + d with itself).

Explanation / Answer

(a) Area of POQ = (1/2) * p X q (coming out of the page) (b) (i) PQ = OQ - OP = q - p. Similarly, PR = r - p, QR = r - q (ii) a = (1/2) * p X q, b = (1/2) * OR X OP = (1/2) * r X p, c = (1/2) * OQ X OR = (1/2) * q X r, d = (1/2) * PR X PQ = (1/2) * (r - p) X (q - p) = (1/2) * (r X q - p X q - r X p + p X p) = (1/2) * (r X q - p X q - r X p + 0) = (1/2) * (r X q - p X q - r X p). Now, a + b + c + d = (1/2) * [(p X q) + (r X p) + (q X r) + (r X q) - (p X q) - (r X p)] =0 (iii) A^2 = [(1/2) * | p X q |]^2 = (1/4) * |p|^2 * |q|^2 * sin 90 = (1/4) * |p|^2 * |q|^2. Similarly, B^2 = (1/4) * |p|^2 * |r|^2 and C^2 = (1/4) * |r|^2 * |q|^2. Hence, A^2 + B^2 + C^2 = (1/4) * [|p|^2.|q|^2 + |p|^2.|r|^2 + |r|^2.|q|^2]. Now, D^2 = (1/4) | (r - p) X (q - p) |^2 = (1/4) | r X q - p X q - r X p |^2 = (1/4) * [ (r X q - p X q - r X p) . (r X q - p X q - r X p) ] = (1/4) * [ |r X q|^2 + |p X q|^2 + |r X p|^2 - 2(r X q).(p X q) - 2(r X q).(r X p) + 2(p X q).(r X p) ] = (1/4) * [ |r|^2.|q|^2 + |p|^2.|q|^2 + |r|^2.|q|^2 ] = A^2 + B^2 + C^2

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