Question 42 of 68intermediate🔧 ApplyNumerical2 marks

Find λ and μ if (2î + 6ĵ + 27k̂) × (î + λĵ + μk̂) = 0⃗.

Correct Answer

λ = 3 and μ = 27/2 (or 13.5)

Exercise: EXERCISE 10.4 | Q: 5 | (Page 31)
For More Understanding

Explanation

This question tests the concept that when the cross product of two non-zero vectors equals zero, the vectors are collinear (parallel). From the textbook context, two vectors are collinear if and only if one is a scalar multiple of the other. The student needs to identify this relationship and solve for the unknown coefficients by comparing corresponding components.

Solution Steps

  1. Step 1: Let vector a = 2î + 6ĵ + 27k̂ and vector b = î + λĵ + μk̂.

  2. Step 2: Since a × b = 0⃗, the two vectors are collinear (parallel).

  3. Step 3: For collinear vectors, there exists a scalar k such that b = k(a).

  4. Step 4: So, î + λĵ + μk̂ = k(2î + 6ĵ + 27k̂)

  5. Step 5: Comparing coefficients of î: 1 = 2k, therefore k = 1/2.

  6. Step 6: Comparing coefficients of ĵ: λ = 6k = 6 × (1/2) = 3.

  7. Step 7: Comparing coefficients of k̂: μ = 27k = 27 × (1/2) = 27/2.

  8. Step 8: Therefore, λ = 3 and μ = 27/2.

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