Find the angle between the following pairs of lines:

Question 13 of 25intermediate🔧 ApplyShort Answer4 marks

(i)

$\vec{r} = 2\hat{i} - 5\hat{j} + \hat{k} + \lambda(3\hat{i} + 2\hat{j} + 6\hat{k})$ and $\vec{r} = 7\hat{i} - 6\hat{k} + \mu(\hat{i} + 2\hat{j} + 2\hat{k})$

Answer

θ = cos⁻¹(19/21)

(ii)

$\vec{r} = 3\hat{i} + \hat{j} - 2\hat{k} + \lambda(\hat{i} - \hat{j} - 2\hat{k})$ and $\vec{r} = 2\hat{i} - \hat{j} - 56\hat{k} + \mu(3\hat{i} - 5\hat{j} - 4\hat{k})$

Answer

θ = cos⁻¹(8 $\sqrt{3}$ /15)

Explanation

This question tests the application of the angle formula between two lines in vector form. The key insight is that the angle between two lines depends only on their direction vectors, not on their position vectors. Students must identify the direction vectors from each line equation and apply the dot product formula correctly.

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