JEE Advance - Mathematics (1997 - No. 5)
If $$A,B$$ and $$C$$ are vectors such that $$\left| B \right| = \left| C \right|.$$ Prove that
$$\left[ {\left( {A + B} \right) \times \left( {A + C} \right)} \right] \times \left( {B \times C} \right)\left( {B + C} \right) = 0\,\,.$$
$$\left[ {\left( {A + B} \right) \times \left( {A + C} \right)} \right] \times \left( {B \times C} \right)\left( {B + C} \right) = 0\,\,.$$
The given equation is always true regardless of the specific vectors A, B, and C.
The equation holds if and only if A, B, and C are coplanar.
The equation holds if and only if B and C are parallel.
The equation holds if and only if A is a scalar multiple of B and C.
The provided equation simplifies to 0 due to properties of the cross product and scalar triple product. The expression involves cross products and a scalar triple product (dot product of a cross product). By expanding and using vector identities, especially the fact that a scalar triple product with two identical vectors is zero, the entire expression simplifies to 0.
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