যদি,vecP=2hati+4hatj-5hatk vecQ=-hati+2hatj+3hatk হলেvecP ও vecQ এর মধ্যবর্তী কোণ কত?

Another Explanation (5): ```html

দুটি ভেক্টরের মধ্যবর্তী কোণ নির্ণয়

ধরি, \( \vec{P} \) ও \( \vec{Q} \) দুইটি ভেক্টর। এদের মধ্যবর্তী কোণ \( \theta \) হলে,

\(\cos\theta = \frac{\vec{P} \cdot \vec{Q}}{|\vec{P}| |\vec{Q}|}\)

\(\vec{P} = 2\hat{i} + 4\hat{j} - 5\hat{k}\)

\(\vec{Q} = -\hat{i} + 2\hat{j} + 3\hat{k}\)

\(\vec{P} \cdot \vec{Q} = (2 \times -1) + (4 \times 2) + (-5 \times 3)\)

\(\vec{P} \cdot \vec{Q} = -2 + 8 - 15 = -9\)

\(|\vec{P}| = \sqrt{2^2 + 4^2 + (-5)^2}\)

\(|\vec{P}| = \sqrt{4 + 16 + 25} = \sqrt{45}\)

\(|\vec{Q}| = \sqrt{(-1)^2 + 2^2 + 3^2}\)

\(|\vec{Q}| = \sqrt{1 + 4 + 9} = \sqrt{14}\)

\(\cos\theta = \frac{-9}{\sqrt{45} \sqrt{14}}\)

\(\cos\theta = \frac{-9}{\sqrt{630}}\)

\(\cos\theta = \frac{-9}{25.0998}\)

\(\cos\theta = -0.35857\)

\(\theta = \cos^{-1}(-0.35857)\)

\(\theta = 111.01^\circ\)

\(\vec{P}\) ও \(\vec{Q}\) এর মধ্যবর্তী কোণ \( 111.01^\circ \)। 🎉

```