Question:
A, B, C, D are any 4 points and
\[
|AB \times CD + BC \times AD + CA \times BD| = \lambda (\text{Area of } \Delta ABC)
\]
then \(\lambda = ?\)
A, B, C, D are any 4 points and
\[
|AB \times CD + BC \times AD + CA \times BD| = \lambda (\text{Area of } \Delta ABC)
\]
then \(\lambda = ?\)
Show Hint
Remember vector identity for four points involving cross products sums to 4 times triangle area.
Updated On: Jul 18, 2026
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The Correct Option is D
Solution and Explanation
Step 1: Recall vector identity.
For any four points A, B, C, D, the sum of these cross products is proportional to the area: \[ AB \times CD + BC \times AD + CA \times BD = 4 (\text{Area of } \Delta ABC) \]
Step 2: Interpret magnitude.
Magnitude of the sum gives the scalar multiple \(\lambda\) of the triangle area.
Step 3: Identify proportionality.
\(\lambda = 4\) by definition of the identity.
Step 4: Verify consistency.
Check dimensions and cross product magnitudes; matches the area scaling.
Step 5: Final conclusion.
\[ \boxed{4} \]
For any four points A, B, C, D, the sum of these cross products is proportional to the area: \[ AB \times CD + BC \times AD + CA \times BD = 4 (\text{Area of } \Delta ABC) \]
Step 2: Interpret magnitude.
Magnitude of the sum gives the scalar multiple \(\lambda\) of the triangle area.
Step 3: Identify proportionality.
\(\lambda = 4\) by definition of the identity.
Step 4: Verify consistency.
Check dimensions and cross product magnitudes; matches the area scaling.
Step 5: Final conclusion.
\[ \boxed{4} \]
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