Question 58 of 68beginner🔧 ApplyNumerical1 mark

Find the unit vector in the direction of the vector a⃗ = î + ĵ + 2k̂.

Correct Answer

16i^+16j^+26k^\frac{1}{\sqrt{6}} \hat{i} + \frac{1}{\sqrt{6}} \hat{j} + \frac{2}{\sqrt{6}} \hat{k}

Exercise: EXERCISE 10.2 | Q: 7 | (Chapter: 17)
For More Understanding

Explanation

This question tests the fundamental concept of finding a unit vector in the direction of a given vector. The solution follows the exact same pattern as Example 6 in the textbook context. The key steps are: (1) Calculate the magnitude of the vector using the formula |a⃗| = x2+y2+z2\sqrt{x² + y² + z²}, and (2) Divide the vector by its magnitude to get the unit vector. The magnitude calculation uses the coefficients 1, 1, and 2 from the vector components.

Solution Steps

  1. Step 1: The unit vector in the direction of a vector a⃗ is given by â = (1/|a⃗|)a⃗.

  2. Step 2: Find the magnitude of the vector a⃗ = î + ĵ + 2k̂.

  3. Step 3: |a⃗| = 12+12+22\sqrt{1² + 1² + 2²} = 1+1+4\sqrt{1 + 1 + 4} = 6\sqrt{6}

  4. Step 4: Therefore, the required unit vector is â = (1/6\sqrt{6})(î + ĵ + 2k̂) = (1/6\sqrt{6})î + (1/6\sqrt{6})ĵ + (2/6\sqrt{6})k̂

PreviousNext