Find the value of b if a unit vector is represented as `(0.8hati+bhatj+sqrt(0.2hatk))`

Explanation

To find the value of \( b \) in the unit vector represented as \( \vec{a} = 0.8 \hat{i} + b \hat{j} + \sqrt{0.2} \hat{k} \), we need to ensure that the magnitude of this vector is equal to 1, since a unit vector has a magnitude of 1.

### Step-by-Step Solution:

1. **Write the expression for the vector:**
\[ \vec{a} = 0.8 \hat{i} + b \hat{j} + \sqrt{0.2} \hat{k} \]

2. **Calculate the magnitude of the vector:**
The magnitude \( |\vec{a}| \) of the vector \( \vec{a} \) is given by:
\[ |\vec{a}| = \sqrt{(0.8)^2 + b^2 + (\sqrt{0.2})^2} \]

3. **Substitute the values:**
Substitute the values into the magnitude formula:
\[ |\vec{a}| = \sqrt{0.64 + b^2 + 0.2} \]

4. **Combine the constants:**
Combine the constants inside the square root:
\[ |\vec{a}| = \sqrt{0.84 + b^2} \]

5. **Set the magnitude equal to 1 (since it's a unit vector):**
\[ \sqrt{0.84 + b^2} = 1 \]

6. **Square both sides to eliminate the square root:**
\[ 0.84 + b^2 = 1 \]

7. **Isolate \( b^2 \):**
\[ b^2 = 1 - 0.84 \]
\[ b^2 = 0.16 \]

8. **Take the square root to find \( b \):**
\[ b = \pm \sqrt{0.16} \]
\[ b = \pm 0.4 \]

Thus, the value of \( b \) can be either \( 0.4 \) or \( -0.4 \).

### Final Answer:
\[ b = 0.4 \text{ or } b = -0.4 \]