Question 21 of 68beginner🔧 ApplyNumerical2 marks

Find the angle between two vectors a⃗ and b⃗ with magnitudes 3\sqrt{3} and 2, respectively having a⃗ ⋅ b⃗ = 6\sqrt{6}.

Correct Answer

The angle between the vectors is π/4.

Exercise: EXERCISE 10.3 | Q: 1 | (Page 24)
For More Understanding

Explanation

The solution follows the method demonstrated in Example 13 of the provided context. It utilizes the standard formula relating the angle between two vectors to their dot product and magnitudes. By substituting the given values into the formula, the cosine of the angle is determined, which is then used to find the angle in radians.

Solution Steps

  1. Step 1: Identify the given values: |a⃗| = 3\sqrt{3}, |b⃗| = 2, and a⃗ ⋅ b⃗ = 6\sqrt{6}.

  2. Step 2: Use the formula for the angle θ between two vectors: θ = cos⁻¹((a⃗ ⋅ b⃗) / (|a⃗| |b⃗|)).

  3. Step 3: Substitute the values into the formula: θ = cos⁻¹(6\sqrt{6} / (√3×23 \times 2)).

  4. Step 4: Simplify the fraction: 6\sqrt{6} / (23\sqrt{3}) = 2\sqrt{2} / 2 = 1/2\sqrt{2}.

  5. Step 5: Calculate the final angle: θ = cos⁻¹(1/2\sqrt{2}) = π/4.

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