SATHEE: Vectors 2 Question 23

Vectors 2 Question 23

23. If $A, B, C, D$ are any four points in space, then prove that $| \vec{A B} \times \vec{C D} + \vec{B C} \times \vec{A D} + \vec{C A} \times \vec{B D} |$

$= 4$ (area of $△ A B C)$ .

(1987, 2M)

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Solution:

Let the position vectors of points $A, B, C, D$ be $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ , respectively.

Then, $\vec{A B} = \vec{b} - \vec{a}, \vec{B C} = \vec{c} - \vec{b}, \vec{A D} = \vec{d} - \vec{a}$ ,

$\vec{B D} = \vec{d} - \vec{b}, \vec{C A} = \vec{a} - \vec{c}, \vec{C D} = \vec{d} - \vec{c}$

Now, $| \vec{A B} \times \vec{C D} + \vec{B C} \times \vec{A D} + \vec{C A} \times \vec{B D} |$

$\begin{aligned} = | (\vec{b} - \vec{a}) \times (\vec{d} - \vec{c}) + (\vec{c} - \vec{b}) \times (\vec{d} - \vec{a}) + (\vec{a} - \vec{c}) \times (\vec{d} - \vec{b}) | \\ =∣ \vec{b} \times \vec{d} - \vec{a} \times \vec{d} - \vec{b} \times \vec{c} + \vec{a} \times \vec{c} + \vec{c} \times \vec{d} - \vec{c} \times \vec{a} - \vec{b} \times \vec{d} \\ + \vec{b} \times \vec{a} + \vec{a} \times \vec{d} - \vec{a} \times \vec{b} - \vec{c} \times \vec{d} + \vec{c} \times \vec{b} ∣ \\ = 2 ∣ \vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}) \end{aligned}$

Also, area of $△ A B C$

$\begin{aligned} = \frac{1}{2} | \vec{A B} \times \vec{A C} | = \frac{1}{2} | (\vec{b} - \vec{a}) \times (\vec{c} - \vec{a}) | \\ = \frac{1}{2} | \vec{b} \times \vec{c} - \vec{b} \times \vec{a} - \vec{a} \times \vec{c} + \vec{a} \times \vec{a} | \\ = \frac{1}{2} | \vec{a} \times \vec{a} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a} | \end{aligned}$

From Eqs. (i) and (ii),

$| \vec{A B} \times \vec{C D} + \vec{B C} \times \vec{A D} + \vec{C A} \times \vec{B D} | 2 (2$ area of $△ A B C)$

$= 4 (area of △ A B C)$

Vectors 2 Question 23

23. If $A, B, C, D$ are any four points in space, then prove that $| \vec{A B} \times \vec{C D} + \vec{B C} \times \vec{A D} + \vec{C A} \times \vec{B D} |$

Solution:

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Vectors 2 Question 23

23. If A,B,C,D are any four points in space, then prove that |AB→×CD→+BC→×AD→+CA→×BD→|

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23. If $A, B, C, D$ are any four points in space, then prove that $| \vec{A B} \times \vec{C D} + \vec{B C} \times \vec{A D} + \vec{C A} \times \vec{B D} |$