Question

If \( |\mathbf{\bar{f}}| = 10\), \( |\mathbf{\bar{g}}| = 14\) and \( |\mathbf{\bar{f}} - \mathbf{\bar{g}}| = 15\), then \( |\mathbf{\bar{f}} + \mathbf{\bar{g}}| =\)

• A. \( 367 \)
• B. \( \sqrt{367} \)
• C. \( 400 \)
• D. \( 20 \)

Hint:

In problems involving vector sums and differences, knowing the magnitude of each and the magnitude of their difference allows you to utilize the properties of dot products to find the magnitude of their sum.

Answer 1

The Correct Option is B

Step 1: Use the vector magnitude properties for sum and difference: \[ |\mathbf{\bar{f}} + \mathbf{\bar{g}}|^2 = |\mathbf{\bar{f}}|^2 + |\mathbf{\bar{g}}|^2 + 2 \mathbf{\bar{f}} \cdot \mathbf{\bar{g}}, \] \[ |\mathbf{\bar{f}} - \mathbf{\bar{g}}|^2 = |\mathbf{\bar{f}}|^2 + |\mathbf{\bar{g}}|^2 - 2 \mathbf{\bar{f}} \cdot \mathbf{\bar{g}}. \] Step 2: Given \( |\mathbf{\bar{f}} - \mathbf{\bar{g}}| = 15 \), substitute and find \( \mathbf{\bar{f}} \cdot \mathbf{\bar{g}} \): \[ 225 = 100 + 196 - 2 \mathbf{\bar{f}} \cdot \mathbf{\bar{g}} \Rightarrow \mathbf{\bar{f}} \cdot \mathbf{\bar{g}} = 35.5. \] Step 3: Calculate \( |\mathbf{\bar{f}} + \mathbf{\bar{g}}|^2 \): \[ |\mathbf{\bar{f}} + \mathbf{\bar{g}}|^2 = 100 + 196 + 71 = 367. \] Step 4: Thus, \[ |\mathbf{\bar{f}} + \mathbf{\bar{g}}| = \sqrt{367}. \]