Question

If \( \mathbf{a}, \mathbf{b}, \mathbf{c} \) are three vectors such that \( |\mathbf{a}| = |\mathbf{b}| = |\mathbf{c}| = \sqrt{3} \) and \( ( \mathbf{a} + \mathbf{b} - \mathbf{c} )^2 + ( \mathbf{b} + \mathbf{c} - \mathbf{a} )^2 + ( \mathbf{c} + \mathbf{a} - \mathbf{b} )^2 = 36\), then \(|2\mathbf{a} - 3\mathbf{b} + 2\mathbf{c}|^2 =\)

• A. \( 15 \)
• B. \( 25 \)
• C. \( 147 \)
• D. \( 75 \)

Hint:

When working with vector equations involving sums and differences, utilizing vector identities and properties like dot products can simplify the process of finding magnitudes.

Answer 1

The Correct Option is D

Step 1: Recognize the vector identity used: \[ |\mathbf{u} + \mathbf{v} + \mathbf{w}|^2 = |\mathbf{u}|^2 + |\mathbf{v}|^2 + |\mathbf{w}|^2 + 2 (\mathbf{u} \cdot \mathbf{v} + \mathbf{v} \cdot \mathbf{w} + \mathbf{w} \cdot \mathbf{u}), \] where \(\mathbf{u}, \mathbf{v}, \mathbf{w}\) are the vectors \(\mathbf{a}, \mathbf{b}, \mathbf{c}\). 
Step 2: Apply this identity to the given vector expressions and rearrange to find relations between dot products: \[ ( \mathbf{a} + \mathbf{b} - \mathbf{c} )^2 + ( \mathbf{b} + \mathbf{c} - \mathbf{a} )^2 + ( \mathbf{c} + \mathbf{a} - \mathbf{b} )^2 = 36, \] which implies simplifications based on the magnitudes and orthogonality of vectors. 
Step 3: Calculate \(|2\mathbf{a} - 3\mathbf{b} + 2\mathbf{c}|^2\) using properties of vector norms and the results obtained from step 2: \[ |2\mathbf{a} - 3\mathbf{b} + 2\mathbf{c}|^2 = 4|\mathbf{a}|^2 + 9|\mathbf{b}|^2 + 4|\mathbf{c}|^2 - 12 \mathbf{a} \cdot \mathbf{b} + 8 \mathbf{a} \cdot \mathbf{c} - 12 \mathbf{b} \cdot \mathbf{c}. \] Plug in known values and solve for the final magnitude.