If a unit vector is represented by `0.5 hat(i) + 0.8 hat(j) + c hat(k)` , then the value of `c` is

Options

A. `1`
B. `sqrt(0.11)`
C. `sqrt(0.01)`
D. `sqrt(0.39)`

Explanation

To find the value of \( c \) in the unit vector represented by \( 0.5 \hat{i} + 0.8 \hat{j} + c \hat{k} \), we need to ensure that the magnitude of this vector is equal to 1, as it is a unit vector.

### Step-by-Step Solution:

1. **Understand the Magnitude of a Vector**:
The magnitude of a vector \( \vec{A} = a \hat{i} + b \hat{j} + c \hat{k} \) is given by the formula:
\[ |\vec{A}| = \sqrt{a^2 + b^2 + c^2} \]

2. **Identify Components**:
In our case, the components are:
- \( a = 0.5 \)
- \( b = 0.8 \)
- \( c = c \)

3. **Set Up the Equation for Magnitude**:
Since it is a unit vector, we have:
\[ \sqrt{(0.5)^2 + (0.8)^2 + c^2} = 1 \]

4. **Square Both Sides**:
To eliminate the square root, square both sides of the equation:
\[ (0.5)^2 + (0.8)^2 + c^2 = 1^2 \]
This simplifies to:
\[ 0.25 + 0.64 + c^2 = 1 \]

5. **Combine the Known Values**:
Now, combine \( 0.25 \) and \( 0.64 \):
\[ 0.89 + c^2 = 1 \]

6. **Isolate \( c^2 \)**:
Subtract \( 0.89 \) from both sides:
\[ c^2 = 1 - 0.89 \]
\[ c^2 = 0.11 \]

7. **Take the Square Root**:
To find \( c \), take the square root of both sides:
\[ c = \sqrt{0.11} \]

8. **Final Value of \( c \)**:
Thus, the value of \( c \) is:
\[ c \approx 0.3317 \]

### Summary:
The value of \( c \) in the unit vector \( 0.5 \hat{i} + 0.8 \hat{j} + c \hat{k} \) is \( \sqrt{0.11} \approx 0.3317 \).