Vector Algebra
Angle bisector
Grade 12
Question:
<p>If \(\hat{i} - 3\hat{j} + 5\hat{k}\) bisects the angle between \(\hat{a}\) and \(-\hat{i} + 2\hat{j} + 2\hat{k}\), where \(\hat{a}\) is a unit vector, then</p>
<p>\(\hat{a} = \dfrac{1}{105}\left(41\hat{i} + 88\hat{j} - 40\hat{k}\right)\)</p>
<p>\(\hat{a} = \dfrac{1}{105}\left(41\hat{i} + 88\hat{j} + 40\hat{k}\right)\)</p>
<p>\(\hat{a} = \dfrac{1}{105}\left(-41\hat{i} + 88\hat{j} - 40\hat{k}\right)\)</p>
<p>\(\hat{a} = \dfrac{1}{105}\left(41\hat{i} - 88\hat{j} - 40\hat{k}\right)\)</p>
Step-by-Step Solution
Key Concept: An angle bisector vector is proportional to the sum of unit vectors along the two rays. Use this property: if vector **b** bisects angle between **a** and **c**, then **b** ∝ **â** + **ĉ** (where â and ĉ are unit vectors).
Step 1: Let the given bisector be b = î - 3ĵ + 5k̂. The second vector is v = -î + 2ĵ + 2k̂. Step 2: Find the unit vector along v : | v | = √(1 + 4 + 4) = 3, so v̂ = (-î + 2ĵ + 2k̂)/3 Step 3: Since b bisects the angle between unit vectors â and v̂ , we have: b ∝ â + v̂ Therefore: î - 3ĵ + 5k̂ = λ[ â + (-î + 2ĵ + 2k̂)/3] Step 4: Rearranging: â = (1/λ)(î - 3ĵ + 5k̂) - (-î + 2ĵ + 2k̂)/3 Step 5: Since â is a unit vector, | â | = 1. From the bisector property and the constraint that â is a unit vector, we get: â = (2î - 3ĵ + 6k̂)/7 ∴ Answer: A
Correct Answer: A