Question
Let $\vec{\text{a}}=2\vec{\text{i}}+3\vec{\text{j}}+4\vec{\text{k}}$ and $\vec{\text{b}}=3\vec{\text{i}}+4\vec{\text{j}}+5\vec{\text{k}}.$ Find the angle between them.
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Answer
$\Rightarrow\vec{\text{a}}=2\vec{\text{i}}+3\vec{\text{j}}+4\vec{\text{k}};\vec{\text{b}}=3\vec{\text{i}}+4\vec{\text{j}}+5\vec{\text{k}}$$\Rightarrow\vec{\text{a}}.\vec{\text{b}}=\text{ ab}\cos\theta\Rightarrow\theta=\cos^{-1}\frac{\vec{\text{a}}.\vec{\text{b}}}{\text{ab}}$
$\Rightarrow\cos^{-1}\frac{2\times3+3\times4+4\times5}{\sqrt{2^2+3^3+4^4}\sqrt{3^2+4^2+5^2}}=\cos^{-1}\Big(\frac{38}{\sqrt{1450}}\Big)$
$\Rightarrow\cos^{-1}\frac{2\times3+3\times4+4\times5}{\sqrt{2^2+3^3+4^4}\sqrt{3^2+4^2+5^2}}=\cos^{-1}\Big(\frac{38}{\sqrt{1450}}\Big)$
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