Q.75. Vectors - application to area of plane figure.

Question 75.
If A, B, C, D be four points in space and if
(AB vector x CD vector) + ( BC vector x AD vector) + (CA vector x BD vector)
= λ*Area of triangle ABC,  then find the value of  λ.

Answer 75.
Let A, B, C, D be denoted by vectors a, b, c and d.
=> l (b-a)x(d-c) + (c-b)x(d-a) + (a-c)x(d-b) l
      = λ * Area of triangle ABC
=> l (bxd) - (bxc) - (axd) + (axc) + (cxd) - (cxa) - (bxd) + (bxa) +
         (axd) - (axb) - (cxd) + (cxb) l
     = λ * Area of triangle ABC

Cancelling (bxd) with -(bxd), -(axd) with (axd) and (cxd) with -(cxd),
=> l -(bxc) + (axc) - (cxa) + (bxa) - (axb) + (cxb) l
      = λ * Area of triangle ABC
=> l 2(cxb) + 2(axc) + 2(bxa) l = λ * Area of triangle ABC
=> 4 * (1/2) l (axb) + (bxc) + (cxa) l = λ * Area of triangle ABC

As (1/2) l (axb) + (bxc) + (cxa) l = area of triangle ABC,
λ = 4.

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