Two adjacent side of parallelogram ABCD are 2-4-5k- and -2-3k- Then the value of - A-C-B-D- - is

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Addition of Vectors

Two adjacent ...

Question

Two adjacent side of parallelogram $A B C D$ are $2^i + 4^j - 5^k$ and $^i + 2^j + 3^k$ . Then the value of $∣ ∣ \to A C \times \to B D ∣ ∣$ is

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Solution

$\begin{matrix} L e t A B C D i s a p a r a l l e log r a m N o w, a r e a o f Δ = \frac{1}{2} (\to a \times \to b) = \frac{1}{2} (- - \to A C \times - - \to B D) \frac{1}{2} ∣ ∣ ∣ - - \to A C \times - - \to B D ∣ ∣ ∣ = \frac{1}{2} ∣ ∣ ∣ ∣ ∣ \begin{matrix} ˆ i & ˆ j & ˆ k 2 & 4 & - 5 1 & 2 & 3 \end{matrix} ∣ ∣ ∣ ∣ ∣ = \frac{1}{2} [ˆ i (12 + 10) - ˆ j (6 + 5) + ˆ k (4 - 4)] = \frac{1}{2} (22 ˆ i - 11 ˆ j) ∴ \frac{1}{2} (- - \to A C \times - - \to B D) = \frac{1}{2} (22 ˆ i - 11 ˆ j) s o, 2 (22 ˆ i - 11 ˆ j) = (- - \to A C \times - - \to B D) = 22 [2 ˆ i - ˆ j] = 22 \sqrt{5} \end{matrix}$

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Two adjacent side of parallelogram ABCD are 2^i+4^j−5^k and ^i+2^j+3^k. Then the value of ∣∣→AC×→BD∣∣ is

Two adjacent side of parallelogram $A B C D$ are $2^i + 4^j - 5^k$ and $^i + 2^j + 3^k$ . Then the value of $∣ ∣ \to A C \times \to B D ∣ ∣$ is