Question

If \( \vec{a} = 3\hat{i} - 2\hat{j} \) and \( \vec{b} = \hat{i} + 4\hat{j} \), find the magnitude of \( 2\vec{a} - 3\vec{b} \).

• A. \( \sqrt{85} \)
• B. \( \sqrt{74} \)
• C. \( \sqrt{265} \)
• D. \( \sqrt{105} \)

Hint:

To find the magnitude of a vector expression like \( a\vec{u} - b\vec{v} \), compute the components separately and then apply the magnitude formula: \[ |\vec{r}| = \sqrt{x^2 + y^2} \]

Answer 1

The Correct Option is C

Step 1: Calculate \( 2\vec{a} \) \[ 2\vec{a} = 2(3\hat{i} - 2\hat{j}) = 6\hat{i} - 4\hat{j} \] Step 2: Calculate \( 3\vec{b} \) \[ 3\vec{b} = 3(\hat{i} + 4\hat{j}) = 3\hat{i} + 12\hat{j} \] Step 3: Compute \( 2\vec{a} - 3\vec{b} \) \[ 2\vec{a} - 3\vec{b} = (6\hat{i} - 4\hat{j}) - (3\hat{i} + 12\hat{j}) = (6 - 3)\hat{i} + (-4 - 12)\hat{j} = 3\hat{i} - 16\hat{j} \] Step 4: Find the magnitude of the resulting vector \[ |\vec{v}| = \sqrt{3^2 + (-16)^2} = \sqrt{9 + 256} = \sqrt{265} \]