`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 \( \mathbf{v} = 0.4 \hat{i} + 0.8 \hat{j} + c \hat{k} \) represents a unit vector, we need to ensure that the magnitude of this vector equals 1.

### Step-by-Step Solution:

1. **Understanding the Magnitude of a Vector**:
The magnitude of a vector \( \mathbf{v} = a \hat{i} + b \hat{j} + c \hat{k} \) is given by the formula:
\[ |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \]
In our case, \( a = 0.4 \), \( b = 0.8 \), and \( c = c \).

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

3. **Squaring Both Sides**:
To eliminate the square root, we square both sides of the equation:
\[ (0.4)^2 + (0.8)^2 + c^2 = 1^2 \]
This simplifies to:
\[ 0.16 + 0.64 + c^2 = 1 \]

4. **Combining Terms**:
Now, combine the constants on the left side:
\[ 0.16 + 0.64 = 0.80 \]
Thus, we have:
\[ 0.80 + c^2 = 1 \]

5. **Isolating \( c^2 \)**:
To find \( c^2 \), we subtract 0.80 from both sides:
\[ c^2 = 1 - 0.80 \]
This gives us:
\[ c^2 = 0.20 \]

6. **Finding \( c \)**:
To find \( c \), we take the square root of both sides:
\[ c = \sqrt{0.20} \]
This can be simplified to:
\[ c = \sqrt{0.2} \]

7. **Considering the Options**:
Since \( c \) can be positive or negative, we typically take the positive root in this context. However, since the question does not specify, we can state:
\[ c = \sqrt{0.2} \quad \text{(which is approximately 0.447)} \]

### Conclusion:
The value of \( c \) that makes the vector a unit vector is:
\[ c = \sqrt{0.2} \]