Question

Let \[ \vec{a} = 2\hat{i} - 3\hat{j} - 5\hat{k}, \quad \vec{b} = 3\hat{i} + 2\hat{j} - 5\hat{k} \] be two vectors and \(\vec{r}\) be a vector in the plane of \(\vec{a} \text{ and } \vec{b}\). If \(\vec{r}\) is orthogonal to \(5\hat{i} - 2\hat{j} + 3\hat{k}\) and the magnitude of \(\vec{r}\) is \(\sqrt{94}\), then \(|\vec{r} \cdot \vec{b}|\) is

• A. \(36\)
• B. \(38\)
• C. \(42\)
• D. \(\mathbf{46}\)

Hint:

When a vector lies in a plane and is orthogonal to a given vector, express it as a linear combination and use dot product constraints.

Answer 1

The Correct Option is D

Since \(\vec{r}\) lies in the plane of \(\vec{a}\) and \(\vec{b}\), write: \[ \vec{r} = m\vec{a} + n\vec{b} \] Given: \[ \vec{r} \cdot (5\hat{i} - 2\hat{j} + 3\hat{k}) = 0 \quad \text{(orthogonal condition)} \] and \[ |\vec{r}| = \sqrt{94} \] Solving this system (not shown due to complexity) leads to: \[ |\vec{r} \cdot \vec{b}| = \boxed{46} \]