Question

If the line joining points $ \vec{r}_1 = \hat{i} + 2\hat{j} $ and $ \vec{r}_2 = \hat{j} - 2\hat{k} $ intersects the plane through the points $ \vec{A} = 2\hat{i} - \hat{j},\ \vec{B} = -2\hat{j} + 3\hat{k},\ \vec{C} = \hat{k} - 2\hat{i} $ at $ T $, then find $ \vec{r}_T \cdot (\hat{i} + \hat{j} + \hat{k}) $

• A. 15
• B. 5
• C. 3
• D. 7

Hint:

Find point of intersection by substituting line into plane and simplify.

Answer 1

The Correct Option is A

Line: \( \vec{r} = \vec{r}_1 + \lambda(\vec{r}_2 - \vec{r}_1) \) Plane: Use 3 points to get normal via cross product Substitute parametric \( \vec{r} \) into plane and solve for \( \lambda \) Find \( \vec{r}_T \) and compute dot product with \( \hat{i} + \hat{j} + \hat{k} \) Final result = 15