The angle between the lines r = 3i + 2j - 4k + (i + 2j + 2k) and r = 5j - 2k + (3i + 2j + 6k) is : Option 1: ^(-1)(19/21) Option 2: ^(-1)(19/23) Option 3: ^(-1)(19/21) Option 4: ^(-1)(19/23)

Answer: ^(-1)(19/21). Solution: Direction vectors are d_1 = (1,2,2) with magnitude 3, and d_2 = (3,2,6) with magnitude 7. Dot product = 3 + 4 + 12 = 19. So = 19/(3 * 7) = 19/21, giving = ^(-1)(19/21).

Geometry

3D Geometry

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The angle between the lines $\vec{r} = 3\hat{i} + 2\hat{j} - 4\hat{k} + \lambda(\hat{i} + 2\hat{j} + 2\hat{k})$ and $\vec{r} = 5\hat{j} - 2\hat{k} + \mu(3\hat{i} + 2\hat{j} + 6\hat{k})$ is :

$\sin^{-1}\left(\frac{19}{21}\right)$

$\cos^{-1}\left(\frac{19}{23}\right)$

$\cos^{-1}\left(\frac{19}{21}\right)$

$\sin^{-1}\left(\frac{19}{23}\right)$

✅ Correct Option: 3

Direction vectors are $\vec{d_1} = (1,2,2)$ with magnitude 3, and $\vec{d_2} = (3,2,6)$ with magnitude 7. Dot product $= 3 + 4 + 12 = 19$ . So $\cos\theta = 19/(3 \cdot 7) = 19/21$ , giving $\theta = \cos^{-1}(19/21)$ .

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