Vector Algebra
Unit Vector
Grade 12
Question:
<p>Given two vectors are <span>\(\hat{i} - \hat{j}\)</span> and <span>\(\hat{i} + 2\hat{j}\)</span> the unit vector coplanar with the two vectors and perpendicular to first is</p>
<p>\(\dfrac{1}{\sqrt{2}}(\hat{i} + \hat{j})\)</p>
<p>\(\dfrac{1}{\sqrt{5}}(2\hat{i} + \hat{j})\)</p>
<p>\(\pm \dfrac{1}{\sqrt{2}}(\hat{i} + \hat{k})\)</p>
<p>none of these</p>
Step-by-Step Solution
Key Concept: A vector perpendicular to the first vector must satisfy the dot product condition equal to zero. To find a unit vector coplanar with both given vectors and perpendicular to the first, construct it as a linear combination of the two vectors, apply perpendicularity constraint, then normalize.
Step 1: Let a = î - ĵ and b = î + 2ĵ. We need a unit vector u coplanar with a and b , and perpendicular to a . Step 2: Since u is coplanar with a and b , write u = λ a + μ b = λ(î - ĵ) + μ(î + 2ĵ) = (λ + μ)î + (-λ + 2μ)ĵ Step 3: For perpendicularity with a : u · a = 0 (λ + μ)(1) + (-λ + 2μ)(-1) = 0 λ + μ + λ - 2μ = 0 2λ - μ = 0 ⟹ μ = 2λ Step 4: Substitute μ = 2λ into u : u = (λ + 2λ)î + (-λ + 4λ)ĵ = 3λî + 3λĵ = 3λ(î + ĵ) Step 5: For unit vector: | u | = 1 |3λ|√(1^2 + 1^2) = 1 3|λ|√2 = 1 ⟹ |λ| = 1/(3√2) Step 6: Therefore u = ±(î + ĵ)/(√2) = ±(î + ĵ)/√2 ∴ Answer: A (The unit vector is (î + ĵ)/√2 or its negative)
Correct Answer: A