Question

The point of intersection of the lines represented by \(\vec{r} = (\hat{i} - 6\hat{j} + 2\hat{k}) + t(\hat{i} + 2\hat{j} + \hat{k})\) and \(\vec{r} = (4\hat{j} + \hat{k}) + s(2\hat{i} + \hat{j} + 2\hat{k})\) is

• A. \(8\hat{i} + 9\hat{j} + 10\hat{k}\)
• B. \(8\hat{i} + 8\hat{j} + 7\hat{k}\)
• C. \(8\hat{i} + 9\hat{j} + 8\hat{k}\)
• D. \(8\hat{i} + 8\hat{j} + 9\hat{k}\)

Hint:

Equate vector parametric forms component-wise to find points of intersection.

Answer 1

The Correct Option is D

Equate the two vector equations: \[ (\hat{i} - 6\hat{j} + 2\hat{k}) + t(\hat{i} + 2\hat{j} + \hat{k}) = (4\hat{j} + \hat{k}) + s(2\hat{i} + \hat{j} + 2\hat{k}) \] Solve component-wise: i-component: \(1 + t = 2s\)
j-component: \(-6 + 2t = 4 + s\)
k-component: \(2 + t = 1 + 2s\)
Solving these gives \(t = 7, s = 4\). Substitute \(t = 7\) into first line: \[ \vec{r} = (\hat{i} - 6\hat{j} + 2\hat{k}) + 7(\hat{i} + 2\hat{j} + \hat{k}) = 8\hat{i} + 8\hat{j} + 9\hat{k}. \]