Given that `0.4hati+0.8hatj+b hatk` is a unit vector . What is the value of b ?

Options

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

Answer

To find the value of \( b \) in the vector \( 0.4 \hat{i} + 0.8 \hat{j} + b \hat{k} \) which is a unit vector, we can follow these steps:

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

### Step 2: Write the expression for the magnitude of the vector
The magnitude of the vector \( \mathbf{v} = 0.4 \hat{i} + 0.8 \hat{j} + b \hat{k} \) can be calculated using the formula:
\[ |\mathbf{v}| = \sqrt{(0.4)^2 + (0.8)^2 + b^2} \]
Since this is a unit vector, we set the magnitude equal to 1:
\[ \sqrt{(0.4)^2 + (0.8)^2 + b^2} = 1 \]

### Step 3: Square both sides to eliminate the square root
Squaring both sides gives:
\[ (0.4)^2 + (0.8)^2 + b^2 = 1^2 \]

### Step 4: Calculate the squares of the components
Calculating \( (0.4)^2 \) and \( (0.8)^2 \):
\[ (0.4)^2 = 0.16 \]
\[ (0.8)^2 = 0.64 \]
Now substitute these values into the equation:
\[ 0.16 + 0.64 + b^2 = 1 \]

### Step 5: Combine the constant terms
Combine \( 0.16 \) and \( 0.64 \):
\[ 0.80 + b^2 = 1 \]

### Step 6: Isolate \( b^2 \)
Subtract \( 0.80 \) from both sides:
\[ b^2 = 1 - 0.80 \]
\[ b^2 = 0.20 \]

### Step 7: Solve for \( b \)
Taking the square root of both sides gives:
\[ b = \pm \sqrt{0.20} \]
\[ b = \pm \sqrt{0.2} = \pm \frac{\sqrt{2}}{5} \]

### Conclusion
The value of \( b \) can be either \( \sqrt{0.2} \) or \( -\sqrt{0.2} \).