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 can follow these steps:

### Step 1: Understand the definition of a unit vector
A unit vector has a magnitude of 1. Therefore, we need to ensure that the magnitude of the vector \( \mathbf{A} = 0.5 \hat{i} + 0.8 \hat{j} + c \hat{k} \) equals 1.

### Step 2: Write the formula for the magnitude of the vector
The magnitude of a vector \( \mathbf{A} = a \hat{i} + b \hat{j} + c \hat{k} \) is given by:
\[ |\mathbf{A}| = \sqrt{a^2 + b^2 + c^2} \]
In our case, \( a = 0.5 \), \( b = 0.8 \), and \( c = c \).

### Step 3: Substitute the values into the magnitude formula
Substituting the values into the formula, we have:
\[ |\mathbf{A}| = \sqrt{(0.5)^2 + (0.8)^2 + c^2} \]

### Step 4: Calculate \( (0.5)^2 \) and \( (0.8)^2 \)
Calculating the squares:
\[ (0.5)^2 = 0.25 \]
\[ (0.8)^2 = 0.64 \]

### Step 5: Set up the equation for the unit vector
Now substituting these values back into the magnitude equation:
\[ |\mathbf{A}| = \sqrt{0.25 + 0.64 + c^2} \]
Since \( |\mathbf{A}| = 1 \), we set up the equation:
\[ \sqrt{0.25 + 0.64 + c^2} = 1 \]

### Step 6: Square both sides to eliminate the square root
Squaring both sides gives:
\[ 0.25 + 0.64 + c^2 = 1 \]

### Step 7: Combine the constants
Combining the constants:
\[ 0.89 + c^2 = 1 \]

### Step 8: Solve for \( c^2 \)
Now, isolate \( c^2 \):
\[ c^2 = 1 - 0.89 \]
\[ c^2 = 0.11 \]

### Step 9: Solve for \( c \)
Taking the square root of both sides gives:
\[ c = \sqrt{0.11} \]

### Conclusion
Thus, the value of \( c \) is \( \sqrt{0.11} \).

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