If `vecA= a hati + 0.5hatj + 0.5hatk` is unit vector, then value of a would be

Options

A. `1/sqrt2`
B. `sqrt2`
C. (1/2)
D. 2

Answer

To find the value of \( a \) such that the vector \( \vec{A} = a \hat{i} + 0.5 \hat{j} + 0.5 \hat{k} \) is a unit vector, we can follow these steps:

### Step 1: Understand the definition of a unit vector
A unit vector is a vector that has a magnitude of 1. Therefore, we need to find the magnitude of the vector \( \vec{A} \) and set it equal to 1.

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

### Step 3: Substitute the values and simplify
Substituting the values of \( b \) and \( c \):
\[ |\vec{A}| = \sqrt{a^2 + 0.25 + 0.25} = \sqrt{a^2 + 0.5} \]

### Step 4: Set the magnitude equal to 1
Since \( \vec{A} \) is a unit vector, we set the magnitude equal to 1:
\[ \sqrt{a^2 + 0.5} = 1 \]

### Step 5: Square both sides to eliminate the square root
Squaring both sides gives:
\[ a^2 + 0.5 = 1 \]

### Step 6: Solve for \( a^2 \)
Rearranging the equation:
\[ a^2 = 1 - 0.5 = 0.5 \]

### Step 7: Take the square root to find \( a \)
Taking the square root of both sides:
\[ a = \pm \sqrt{0.5} = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2} \]

### Conclusion
The value of \( a \) can be either \( \frac{1}{\sqrt{2}} \) or \( -\frac{1}{\sqrt{2}} \).

---