`0.4hat(i)+0.8hat(j)+chat(k)` represents a unit vector, when c is :-

Options

A. 0.2
B. `sqrt(0.2)`
C. `sqrt(0.8)`
D. 0

Explanation

To determine the value of \( c \) such that the vector \( \vec{v} = 0.4 \hat{i} + 0.8 \hat{j} + c \hat{k} \) represents 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.

### Step 2: Write the expression for the magnitude of the vector
The magnitude \( |\vec{v}| \) of the vector \( \vec{v} \) can be calculated using the formula:
\[ |\vec{v}| = \sqrt{(0.4)^2 + (0.8)^2 + c^2} \]

### Step 3: Set the magnitude equal to 1
Since we want the vector to be a unit vector, we set the magnitude equal to 1:
\[ \sqrt{(0.4)^2 + (0.8)^2 + c^2} = 1 \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (0.4)^2 + (0.8)^2 + c^2 = 1^2 \]
This simplifies to:
\[ 0.16 + 0.64 + c^2 = 1 \]

### Step 5: Combine the constant terms
Adding the constant terms on the left side:
\[ 0.8 + c^2 = 1 \]

### Step 6: Solve for \( c^2 \)
Now, isolate \( c^2 \):
\[ c^2 = 1 - 0.8 \]
\[ c^2 = 0.2 \]

### Step 7: Take the square root to find \( c \)
Taking the square root of both sides gives:
\[ c = \sqrt{0.2} \]
Thus, \( c \) can be either \( \sqrt{0.2} \) or \( -\sqrt{0.2} \).

### Final Answer
The value of \( c \) that makes the vector a unit vector is:
\[ c = \sqrt{0.2} \quad \text{or} \quad c = -\sqrt{0.2} \]

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