Find an expression for the sine of the angle between the two vectors $3^i +^j + 2^k$ and $2^i - 2^j + 4^k$ .

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Solution

Let $\to a = 3^i +^j + 2^k$ and $\to b = 2^i - 2^j + 4^k$ .
$| \to a | = \sqrt{3^{2} + 1^{2} + 2^{2}} = \sqrt{14}$
$| \to b | = \sqrt{2^{2} + 2^{2} + 4^{2}} = \sqrt{24}$
$\to a \times \to b = | \to a | | \to b | sin θ$ ............. $(1)$
Now, $\to a \times \to b = ⎡ ⎢ ⎣ \begin{matrix} ^i & ^j & ^k 3 & 1 & 2 2 & - 2 & 4 \end{matrix} ⎤ ⎥ ⎦$
$= (4 + 4)^i - (12 - 4)^j + (- 6 - 2)^k$
$= 8^i - 8^j - 8^k$
$∴$ From $(1)$ ,
$sin θ = \frac{8^i - 8^i - 8^k}{\sqrt{14} \sqrt{24}} = \frac{8 (^i -^j -^k)}{\sqrt{2 \times 7 \times 2 \times 2 \times 2 \times 3}}$
$= \frac{8 (^i -^j -^k)}{4 \sqrt{21}}$
$sin θ = \frac{2}{\sqrt{21}} (^i -^j -^k)$ Ans

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Find the sine of the angle between the vectors $\to a = 3^i +^j + 2^k a n d \to b = 2^i - 2^j + 4^k$ .

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3^i +^j + 2^k

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