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Dot Product of Two Vectors

Let a=î+αĵ+3k...

Question

Let $\to a =^i + α^j + 3^k$ and $\to b = 3^i - α^j +^k$ . If the area of the parallelogram whose adjacent sides are represented by the vectors $\to a$ and $\to b$ is $8 \sqrt{3}$ square units, then $\to a \cdot \to b$ is equal to

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Solution

$\to a =^i + α^j + 3^k$
$\to b = 3^i - α^j +^k$
Area of parallelogram $= ∣ ∣ ∣ \to a \times \to b ∣ ∣ ∣$
$= | (^i + α^j + 3^k) \times (3^i - α^j +^k) |$
$\Rightarrow 8 \sqrt{3} = | (4 α)^i + 8^j - (4 α)^k |$
$\Rightarrow (64) (3) = 16 α^{2} + 64 + 16 α^{2}$
$\Rightarrow α^{2} = 4$
Now, $\to a \cdot \to b = 3 - α^{2} + 3$
$= 6 - α^{2}$
$= 6 - 4$
$= 2$

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