If `0.5hati + 0.8hatj + chatk` is aunit vector then c is

Options

A. `sqrt(0.89)`
B. `0.2`
C. `0.3`
D. `sqrt(0.11)`

Explanation

To find the value of \( c \) such that the vector \( \mathbf{v} = 0.5\hat{i} + 0.8\hat{j} + c\hat{k} \) is a unit vector, we need to follow these steps:

### Step 1: Understand the condition for a unit vector
A vector is considered a unit vector if its magnitude is equal to 1. Therefore, we need to calculate the magnitude of the vector \( \mathbf{v} \) and set it equal to 1.

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

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

### Step 3: Set the magnitude equal to 1
Since we want \( |\mathbf{v}| = 1 \), we have:
\[ \sqrt{(0.5)^2 + (0.8)^2 + c^2} = 1 \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (0.5)^2 + (0.8)^2 + c^2 = 1^2 \]
This simplifies to:
\[ 0.25 + 0.64 + c^2 = 1 \]

### Step 5: Combine the constants
Adding \( 0.25 \) and \( 0.64 \):
\[ 0.89 + c^2 = 1 \]

### Step 6: Isolate \( c^2 \)
Subtract \( 0.89 \) from both sides:
\[ c^2 = 1 - 0.89 \]
This simplifies to:
\[ c^2 = 0.11 \]

### Step 7: Solve for \( c \)
Taking the square root of both sides:
\[ c = \sqrt{0.11} \]
Thus, the value of \( c \) is:
\[ c \approx 0.3317 \]

### Final Answer
Therefore, the value of \( c \) is approximately \( 0.3317 \).

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