Find |a⃗| and |b⃗|, if (a⃗ + b⃗) ⋅ (a⃗ - b⃗) = 8 and |a⃗| = 8 |b⃗|.
|a⃗| = 16/21 and |b⃗| = 2/21
Explanation
This question tests the property of dot product expansion. Following the pattern shown in Example 17 and Example 18 from the textbook, we expand (a⃗ + b⃗) ⋅ (a⃗ - b⃗) using the distributive property. The key insight is that this simplifies to |a⃗|² - |b⃗|², which creates a system of equations when combined with the given relationship |a⃗| = 8|b⃗|.
Solution Steps
Step 1: Expand the dot product: (a⃗ + b⃗) ⋅ (a⃗ - b⃗) = a⃗ ⋅ a⃗ - a⃗ ⋅ b⃗ + b⃗ ⋅ a⃗ - b⃗ ⋅ b⃗ = |a⃗|² - |b⃗|²
Step 2: Given that (a⃗ + b⃗) ⋅ (a⃗ - b⃗) = 8, we get: |a⃗|² - |b⃗|² = 8
Step 3: Given |a⃗| = 8|b⃗|, let |b⃗| = x, then |a⃗| = 8x
Step 4: Substitute into the equation: (8x)² - x² = 8
Step 5: Simplify: 64x² - x² = 8 → 63x² = 8 → x² = 8/63
Step 6: Solve: x = = 2/21
Step 7: Therefore, |b⃗| = 2/21 and |a⃗| = $$8 \times 2$$$\sqrt{14}\sqrt{14}$/21