Question 40 of 68intermediate🔧 ApplyNumerical2 marks

If a unit vector a⃗ makes angles π/3 with î, π/4 with ĵ and an acute angle θ with k̂, then find θ and hence, the components of a⃗.

Correct Answer

θ = π/3 and the components of a⃗ are (1/2, 1/2\sqrt{2}, 1/2), so a⃗ = (1/2)î + (1/2\sqrt{2})ĵ + (1/2)k̂

Exercise: EXERCISE 10.4 | Q: 3 | (Page 31)
For More Understanding

Explanation

This question tests the fundamental property of direction cosines. For any vector with direction angles α, β, γ with the coordinate axes, the direction cosines satisfy the relation cos²α + cos²β + cos²γ = 1. Since a⃗ is a unit vector, its components are exactly equal to its direction cosines. The acute angle condition ensures we take the positive value of cos θ.

Solution Steps

  1. Step 1: Let α = π/3, β = π/4, γ = θ be the direction angles of unit vector a⃗ with î, ĵ, k̂ respectively.

  2. Step 2: Using the property of direction cosines: cos²α + cos²β + cos²γ = 1

  3. Step 3: Substituting values: cos²(π/3) + cos²(π/4) + cos²θ = 1

  4. Step 4: This gives: (1/2)² + (1/2\sqrt{2})² + cos²θ = 1, which simplifies to: 1/4 + 1/2 + cos²θ = 1

  5. Step 5: Solving: cos²θ = 1/4, so cos θ = ±1/2. Since θ is acute, cos θ = 1/2, hence θ = π/3.

  6. Step 6: For a unit vector, components equal direction cosines: a⃗ = cos(π/3)î + cos(π/4)ĵ + cos(π/3)k̂ = (1/2)î + (1/2\sqrt{2})ĵ + (1/2)k̂

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