Question

Let \[ \vec{a} = 2\hat{i} + \hat{j} - \hat{k} \quad \text{and} \quad \vec{b} = \hat{i} + 3\hat{j} - 5\hat{k} \] be two vectors, and \(\vec{r}\) be a vector along the vector \(3\vec{a} - 2\vec{b}\) such that \(|\vec{r}| = \sqrt{74}\). If the direction of \(\vec{r}\) is opposite to that of \(3\vec{a} - 2\vec{b}\), then \(\vec{r} =\)

• A. \(-7\hat{i} - 4\hat{j} + 3\hat{k}\)
• B. \(4\hat{i} + 7\hat{j} - 3\hat{k}\)
• C. \(\mathbf{-4\hat{i} + 3\hat{j} - 7\hat{k}}\)
• D. \(4\hat{i} - 3\hat{j} + 7\hat{k}\)

Hint:

When a vector is in the opposite direction, multiply by \(-1\). Then scale to match the given magnitude.

Answer 1

The Correct Option is C

We are given: \[ \vec{a} = 2\hat{i} + \hat{j} - \hat{k}, \quad \vec{b} = \hat{i} + 3\hat{j} - 5\hat{k} \] Compute: \[ 3\vec{a} = 6\hat{i} + 3\hat{j} - 3\hat{k}, \quad 2\vec{b} = 2\hat{i} + 6\hat{j} - 10\hat{k} \] \[ 3\vec{a} - 2\vec{b} = (6 - 2)\hat{i} + (3 - 6)\hat{j} + (-3 + 10)\hat{k} = 4\hat{i} - 3\hat{j} + 7\hat{k} \] Now, since the direction of \(\vec{r}\) is **opposite**, we take: \[ \vec{r} = -k(3\vec{a} - 2\vec{b}) = -k(4\hat{i} - 3\hat{j} + 7\hat{k}) \] Given \(|\vec{r}| = \sqrt{74}\), we compute: \[ | -k(4\hat{i} - 3\hat{j} + 7\hat{k}) | = \sqrt{74} \Rightarrow k \sqrt{(4)^2 + (-3)^2 + (7)^2} = \sqrt{74} \Rightarrow k \sqrt{74} = \sqrt{74} \Rightarrow k = 1 \] Hence, \[ \vec{r} = -1(4\hat{i} - 3\hat{j} + 7\hat{k}) = -4\hat{i} + 3\hat{j} - 7\hat{k} \]