<p>Let <span class="latex">\(\vec{m}\)</span> and <span class="latex">\(\vec{n}\)</span> are unit vectors and <span class="latex">\(\vec{w}\)</span> is vector such that <span class="latex">\(\vec{m} \times \vec{n} + \vec{m} = \vec{w}\)</span> and <span class="latex">\(\vec{w} \times \vec{m} = \vec{n}\)</span>. Then find the value of <span class="latex">\([\vec{m}, \vec{n}, \vec{w}]\)</span>.</p>
Step-by-Step Solution
Key Concept: Use the given vector equations to establish relationships between vectors, then apply the scalar triple product properties and the constraint that m and n are unit vectors to find [m, n, w].
Step 1: Write the given conditions. Given: (i) m × n + m = w, (ii) w × m = n, where |m| = |n| = 1 Step 2: Find m · n using the second condition. From w × m = n, taking dot product with m: (w × m) · m = n · m Since (w × m) · m = 0 (scalar triple product with repeated vector), we get: n · m = 0 Therefore, m ⊥ n. Step 3: Express w using the first condition. From m × n + m = w, we have: w = m × n + m Step 4: Verify the second condition. w × m = (m × n + m) × m = (m × n) × m + m × m Using (a × b) × c = (a · c)b - (b · c)a: (m × n) × m = (m · m)n - (n · m)m = n - 0 = n ✓ This confirms our expression for w is correct. Step 5: Calculate the scalar triple product [m, n, w]. [m, n, w] = m · (n × w) = m · [n × (m × n + m)] = m · [n × (m × n) + n × m] Using (a × b) × c = (a · c)b - (b · c)a, so n × (m × n) = (n · n)m - (n · m)n = m - 0 = m = m · [m + n × m] = m · m + m · (n × m) = 1 + 0 = 1 (Since m · m = 1 as m is a unit vector, and m · (n × m) = 0 as the scalar triple product with a repeated vector is zero) ∴ Answer: 1
Correct Answer: 1