Question

If \( |\mathbf{a}| = 2 \), \( |\mathbf{b}| = 3 \) and \( \mathbf{a} \cdot \mathbf{b} = 4 \), then evaluate \( |\mathbf{a} + 2\mathbf{b}| \).

Hint:

To compute the magnitude of a vector sum, use the formula: \[ |\mathbf{A} + \mathbf{B}|^2 = |\mathbf{A}|^2 + |\mathbf{B}|^2 + 2 (\mathbf{A} \cdot \mathbf{B}). \]

Answer 1

We use the formula for the magnitude of the sum of vectors: \[ |\mathbf{A} + \mathbf{B}|^2 = |\mathbf{A}|^2 + |\mathbf{B}|^2 + 2 (\mathbf{A} \cdot \mathbf{B}). \] Define: \[ \mathbf{A} = \mathbf{a}, \quad \mathbf{B} = 2\mathbf{b}. \] Step 1: Compute \( |\mathbf{B}| \). \[ |\mathbf{B}| = |2\mathbf{b}| = 2 |\mathbf{b}| = 2(3) = 6. \] Step 2: Compute \( \mathbf{A} \cdot \mathbf{B} \). \[ \mathbf{A} \cdot \mathbf{B} = \mathbf{a} \cdot (2\mathbf{b}) = 2 (\mathbf{a} \cdot \mathbf{b}) = 2(4) = 8. \] Step 3: Compute \( |\mathbf{A} + \mathbf{B}| \). \[ |\mathbf{a} + 2\mathbf{b}|^2 = |\mathbf{a}|^2 + |\mathbf{B}|^2 + 2 (\mathbf{a} \cdot \mathbf{B}). \] Substituting values: \[ |\mathbf{a} + 2\mathbf{b}|^2 = 2^2 + 6^2 + 2(8). \] \[ = 4 + 36 + 16 = 56. \] Step 4: Take the square root. \[ |\mathbf{a} + 2\mathbf{b}| = \sqrt{56} = 2\sqrt{14}. \] Thus, the final answer is: \[ |\mathbf{a} + 2\mathbf{b}| = 2\sqrt{14}. \]