JEE Advance - Mathematics (1987 - No. 4)
If $$A, B, C, D$$ are any four points in space, prove that -
$$\left| {\overrightarrow {AB} \times \overrightarrow {CD} + \overrightarrow {BC} \times \overrightarrow {AD} + \overrightarrow {CA} \times \overrightarrow {BD} } \right| = 4$$ (area of triangle $$ABC$$)
$$\left| {\overrightarrow {AB} \times \overrightarrow {CD} + \overrightarrow {BC} \times \overrightarrow {AD} + \overrightarrow {CA} \times \overrightarrow {BD} } \right| = 4$$ (area of triangle $$ABC$$)
The given expression represents 6 times the volume of the tetrahedron ABCD.
The expression simplifies to $$\left| {\overrightarrow {AB} \times \overrightarrow {CD} + \overrightarrow {BC} \times \overrightarrow {AD} + \overrightarrow {CA} \times \overrightarrow {BD} } \right| = 2$$ (area of triangle $$ABC$$)
The expression $$overrightarrow {AB} \times \overrightarrow {CD} + \overrightarrow {BC} \times \overrightarrow {AD} + \overrightarrow {CA} \times \overrightarrow {BD}$$ is always equal to the zero vector.
The given expression equals 4 times the area of triangle ABC only if D lies in the plane of ABC.
The expression simplifies to $$\left| {\overrightarrow {AB} \times \overrightarrow {AC} } \right|$$, which is twice the area of triangle ABC.
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