Find the value of `lambda` in the unit vector `0.4 hati + 0.8 hatj + lambda hatk`.

Answer

To find the value of \( \lambda \) in the unit vector \( \mathbf{v} = 0.4 \hat{i} + 0.8 \hat{j} + \lambda \hat{k} \), we need to ensure that the magnitude of this vector equals 1, since it is a unit vector.

### Step-by-Step Solution:

1. **Understand the Magnitude of a Vector**:
The magnitude of a vector \( \mathbf{v} = a \hat{i} + b \hat{j} + c \hat{k} \) is given by the formula:
\[ |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \]

2. **Apply the Formula to Our Vector**:
For our vector \( \mathbf{v} = 0.4 \hat{i} + 0.8 \hat{j} + \lambda \hat{k} \), we have:
\[ |\mathbf{v}| = \sqrt{(0.4)^2 + (0.8)^2 + \lambda^2} \]

3. **Set the Magnitude Equal to 1**:
Since \( \mathbf{v} \) is a unit vector, we set its magnitude equal to 1:
\[ \sqrt{(0.4)^2 + (0.8)^2 + \lambda^2} = 1 \]

4. **Square Both Sides**:
To eliminate the square root, we square both sides:
\[ (0.4)^2 + (0.8)^2 + \lambda^2 = 1^2 \]

5. **Calculate the Squares**:
Calculate \( (0.4)^2 \) and \( (0.8)^2 \):
\[ (0.4)^2 = 0.16 \quad \text{and} \quad (0.8)^2 = 0.64 \]

6. **Substitute the Values**:
Substitute these values into the equation:
\[ 0.16 + 0.64 + \lambda^2 = 1 \]

7. **Combine Like Terms**:
Combine \( 0.16 \) and \( 0.64 \):
\[ 0.8 + \lambda^2 = 1 \]

8. **Isolate \( \lambda^2 \)**:
Subtract \( 0.8 \) from both sides:
\[ \lambda^2 = 1 - 0.8 \]

9. **Calculate \( \lambda^2 \)**:
This simplifies to:
\[ \lambda^2 = 0.2 \]

10. **Find \( \lambda \)**:
Take the square root of both sides to find \( \lambda \):
\[ \lambda = \sqrt{0.2} \quad \text{or} \quad \lambda = -\sqrt{0.2} \]

11. **Final Values of \( \lambda \)**:
Thus, the possible values of \( \lambda \) are:
\[ \lambda = 0.447 \quad \text{or} \quad \lambda = -0.447 \]