If A, B, C, D are any four points in...

MathematicsVector (Cross) ProductJEE Advanced 1987Moderate

If $A, B, C, D$ are any four points in space, prove that $∣ A B \times C D + B C \times A D + C A \times B D ∣ = 4 (area of triangle A B C)$ .

Visualized Solution (Hindi)

Defining Position Vectors

Let the position vectors of points $A, B, C, D$ be $a, b, c$ , and $d$ respectively.
Any vector $P Q$ can be written as $q - p$ .

Expressing Segments in Vector Form

$A B = b - a$ , $C D = d - c$
$B C = c - b$ , $A D = d - a$
$C A = a - c$ , $B D = d - b$

Substituting into the Expression

Expression: $∣ A B \times C D + B C \times A D + C A \times B D ∣$
Substituting: $∣ (b - a) \times (d - c) + (c - b) \times (d - a) + (a - c) \times (d - b) ∣$

Expanding the First Term

First Term: $(b - a) \times (d - c)$
$= b \times d - b \times c - a \times d + a \times c$

Expanding the Second Term

Second Term: $(c - b) \times (d - a)$
$= c \times d - c \times a - b \times d + b \times a$

Expanding the Third Term

Third Term: $(a - c) \times (d - b)$
$= a \times d - a \times b - c \times d + c \times b$

Summation and Cancellation

Summing all terms, we see cancellations:
$(b \times d - b \times d) = 0$ , $(a \times d - a \times d) = 0$ , $(c \times d - c \times d) = 0$
Remaining: $- b \times c + a \times c - c \times a + b \times a - a \times b + c \times b$
Using $x \times y = - y \times x$ , the sum becomes:
$= 2∣ b \times a + c \times b + a \times c ∣$

Area of Triangle .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A B C

Area of $Δ A B C = \frac{1}{2} ∣ B C \times B A ∣$
$= \frac{1}{2} ∣ (c - b) \times (a - b) ∣$
$= \frac{1}{2} ∣ c \times a - c \times b - b \times a + b \times b ∣$
$= \frac{1}{2} ∣ b \times a + c \times b + a \times c ∣$

Final Conclusion

From step 7: Sum $= 2∣ b \times a + c \times b + a \times c ∣$
From step 8: $∣ b \times a + c \times b + a \times c ∣ = 2 \times Area (Δ A B C)$
Therefore, Sum $= 2 \times (2 \times Area (Δ A B C)) = 4 \times Area (Δ A B C)$
Key Takeaway: The cyclic sum of cross products of opposite edges of a tetrahedron relates directly to the face area.

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a

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If A,B,C,D are any four points in space, prove that ∣AB×CD+BC×AD+CA×BD∣=4(area of triangle ABC).

Visualized Solution (Hindi)

Defining Position Vectors

Expressing Segments in Vector Form

Substituting into the Expression

Expanding the First Term

Expanding the Second Term

Expanding the Third Term

Summation and Cancellation

Area of Triangle .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ABC

Final Conclusion

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