If a unit vector represented by `0.5hati-0.8hatj+chatk`, then the value of 'c' is :

Options

A. torque and work
B. momentum and planck's constant
C. stress and Young's modulus
D. speed and `(mu_(0) in_(0))^(-.^(1)//_(2))`

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 will follow these steps:

### Step 1: Understand the definition of a unit vector
A unit vector is defined as a vector whose magnitude is equal to 1.

### Step 2: Write the expression for the magnitude of the vector
The magnitude of the vector \( \vec{v} = 0.5 \hat{i} - 0.8 \hat{j} + c \hat{k} \) can be calculated using the formula:
\[ |\vec{v}| = \sqrt{(0.5)^2 + (-0.8)^2 + c^2} \]

### Step 3: Set the magnitude equal to 1
Since we want this vector to be a unit vector, we set the magnitude equal to 1:
\[ \sqrt{(0.5)^2 + (-0.8)^2 + c^2} = 1 \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (0.5)^2 + (-0.8)^2 + c^2 = 1 \]

### Step 5: Calculate the squares of the components
Calculating the squares:
\[ (0.5)^2 = 0.25 \quad \text{and} \quad (-0.8)^2 = 0.64 \]
Thus, we have:
\[ 0.25 + 0.64 + c^2 = 1 \]

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

### Step 7: Solve for \(c^2\)
Subtract 0.89 from both sides:
\[ c^2 = 1 - 0.89 = 0.11 \]

### 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:
\[ c = \pm \sqrt{0.11} \]