If a unit vector is represented by `0.5 hat(i)-0.8 hat(j)+chat(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 \( \mathbf{r} = 0.5 \hat{i} - 0.8 \hat{j} + c \hat{k} \), we will follow these steps:

### Step 1: Understand the Definition of a Unit Vector
A unit vector has a magnitude of 1. Therefore, we need to calculate the magnitude of the vector \( \mathbf{r} \) and set it equal to 1.

### Step 2: Calculate the Magnitude of the Vector
The magnitude of a vector \( \mathbf{r} = a \hat{i} + b \hat{j} + c \hat{k} \) is given by the formula:
\[ |\mathbf{r}| = \sqrt{a^2 + b^2 + c^2} \]
For our vector \( \mathbf{r} = 0.5 \hat{i} - 0.8 \hat{j} + c \hat{k} \), we have:
- \( a = 0.5 \)
- \( b = -0.8 \)
- \( c = c \)

Thus, the magnitude becomes:
\[ |\mathbf{r}| = \sqrt{(0.5)^2 + (-0.8)^2 + c^2} \]

### Step 3: Substitute the Values
Now, substituting the values into the magnitude formula:
\[ |\mathbf{r}| = \sqrt{0.25 + 0.64 + c^2} \]

### Step 4: Set the Magnitude Equal to 1
Since \( \mathbf{r} \) is a unit vector, we set the magnitude equal to 1:
\[ \sqrt{0.25 + 0.64 + c^2} = 1 \]

### Step 5: Square Both Sides
To eliminate the square root, we square both sides:
\[ 0.25 + 0.64 + c^2 = 1 \]

### Step 6: Simplify the Equation
Combine the constants:
\[ 0.89 + c^2 = 1 \]

### Step 7: Solve for \( c^2 \)
Now, isolate \( c^2 \):
\[ c^2 = 1 - 0.89 \]
\[ c^2 = 0.11 \]

### Step 8: Find the Value of \( c \)
Taking the square root of both sides gives:
\[ c = \pm \sqrt{0.11} \]

### Final Answer
Thus, the value of \( c \) is:
\[ c = \pm \sqrt{0.11} \]