The two adjacent side of a parallelogram are 2-4-5k- and -2-3k- Find the unit vector parallel to its diagonal- Also find its area-

a⃗ =2î 4ĵ +5k̂ ;b⃗ =î 2ĵ 3k̂ d⃗ ⃗_⃗ ⃗1⃗ =a⃗ +b⃗ =3î 6ĵ +2k̂ d⃗ ⃗_⃗ ⃗2⃗ =a⃗ b⃗ =î 2ĵ +8k̂ d̂ ̂_̂ ̂1̂ = 1 7 ( 3î 6ĵ +2k̂ #160; ) ;d̂ ...

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Standard XII

Mathematics

Dot Product of Two Vectors

The two adjac...

Question

The two adjacent side of a parallelogram are $2^i - 4^j + 5^k$ and $^i - 2^j - 3^k$ . Find the unit vector parallel to its diagonal. Also find its area.

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Solution

$\to a = 2^i - 4^j + 5^k; \to b =^i - 2^j - 3^k$
$_{1} = \to a + \to b = 3^i - 6^j + 2^k$
$_{2} = \to a - \to b =^i - 2^j + 8^k$
$_{1} = \frac{1}{7} (3^i - 6^j + 2^k);_{2} = \frac{1}{\sqrt{69}} (^i - 2^j + 8^k)$
Area $= \frac{1}{2} ∣ ∣_{1} \times_{2} ∣ ∣$
$= \frac{1}{2} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ \begin{matrix} ^i & ^j & ^k 3 & - 6 & 2 1 & - 2 & 8 \end{matrix} ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ = \frac{1}{2} ∣ ∣ - 44^i - 22^j +^k ∣ ∣$
$= \frac{22}{2} ∣ ∣ \sqrt{4 + 1} ∣ ∣ = 11 \sqrt{5} {u n i t}^{2}$

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The two adjacent side of a parallelogram are 2^i−4^j+5^k and ^i−2^j−3^k. Find the unit vector parallel to its diagonal. Also find its area.

The two adjacent side of a parallelogram are $2^i - 4^j + 5^k$ and $^i - 2^j - 3^k$ . Find the unit vector parallel to its diagonal. Also find its area.