`0.4 hat(i)+0.8 hat(j) + c hat(k)` represents a unit vector when c is :

Options

A. 0.2
B. `sqrt(0.2)`
C. `sqrt(0.8)`
D. 0

Answer

To determine the value of \( c \) such that the vector \( 0.4 \hat{i} + 0.8 \hat{j} + c \hat{k} \) represents a unit vector, we need to follow these steps:

### Step-by-Step Solution:

1. **Understanding Unit Vectors**:
A unit vector is defined as a vector with a magnitude of 1.

2. **Magnitude of the Vector**:
The magnitude of the vector \( \mathbf{v} = 0.4 \hat{i} + 0.8 \hat{j} + c \hat{k} \) can be calculated using the formula:
\[ |\mathbf{v}| = \sqrt{(0.4)^2 + (0.8)^2 + c^2} \]

3. **Setting the Magnitude Equal to 1**:
Since we want this vector to be a unit vector, we set the magnitude equal to 1:
\[ \sqrt{(0.4)^2 + (0.8)^2 + c^2} = 1 \]

4. **Squaring Both Sides**:
To eliminate the square root, we square both sides:
\[ (0.4)^2 + (0.8)^2 + c^2 = 1^2 \]

5. **Calculating the Squares**:
Now, calculate \( (0.4)^2 \) and \( (0.8)^2 \):
\[ (0.4)^2 = 0.16 \quad \text{and} \quad (0.8)^2 = 0.64 \]

6. **Substituting the Values**:
Substitute these values back into the equation:
\[ 0.16 + 0.64 + c^2 = 1 \]

7. **Combining Like Terms**:
Combine \( 0.16 \) and \( 0.64 \):
\[ 0.8 + c^2 = 1 \]

8. **Isolating \( c^2 \)**:
Subtract \( 0.8 \) from both sides:
\[ c^2 = 1 - 0.8 \]
\[ c^2 = 0.2 \]

9. **Taking the Square Root**:
Finally, take the square root of both sides to find \( c \):
\[ c = \sqrt{0.2} \]

### Final Answer:
Thus, the value of \( c \) is:
\[ c = \sqrt{0.2} \]