Question 56 of 68intermediate🔧 ApplyNumerical2 marks

If a⃗ = î + ĵ + k̂, b⃗ = 2î - ĵ + 3k̂ and c⃗ = î - 2ĵ + k̂, find a unit vector parallel to the vector 2a⃗ - b⃗ + 3c⃗.

Correct Answer

The unit vector parallel to 2a⃗ - b⃗ + 3c⃗ is (3/22\sqrt{22})î - (3/22\sqrt{22})ĵ + (2/22\sqrt{22})k̂.

Exercise: MISCELLANEOUS EXERCISE ON CHAPTER 10 | Q: 7 | (Page 35)
For More Understanding

Explanation

This question tests the concept of finding a unit vector in the direction of a given vector. The textbook context (Example 7 and Example 8) demonstrates the method: first compute the resultant vector, then find its magnitude, and finally divide the vector by its magnitude. The formula for unit vector is â = (1/|a⃗|)a⃗.

Solution Steps

  1. Step 1: Compute 2a⃗ = 2(î + ĵ + k̂) = 2î + 2ĵ + 2k̂

  2. Step 2: Compute 3c⃗ = 3(î - 2ĵ + k̂) = 3î - 6ĵ + 3k̂

  3. Step 3: Find the resultant vector: 2a⃗ - b⃗ + 3c⃗ = (2î + 2ĵ + 2k̂) - (2î - ĵ + 3k̂) + (3î - 6ĵ + 3k̂) = 3î - 3ĵ + 2k̂

  4. Step 4: Calculate magnitude: |3î - 3ĵ + 2k̂| = 9+9+4\sqrt{9 + 9 + 4} = 22\sqrt{22}

  5. Step 5: The required unit vector = (3î - 3ĵ + 2k̂)/22\sqrt{22} = (3/22\sqrt{22})î - (3/22\sqrt{22})ĵ + (2/22\sqrt{22})k̂

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