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 has a magnitude of 1. Therefore, we need to ensure that the magnitude of \( \vec{A} \) equals 1.

### Step 2: Write the expression for the magnitude of the vector
The magnitude of the vector \( \vec{A} \) can be calculated using the formula:
\[ |\vec{A}| = \sqrt{(a)^2 + (0.5)^2 + (0.5)^2} \]

### Step 3: Set the magnitude equal to 1
Since \( \vec{A} \) is a unit vector, we set the magnitude equal to 1:
\[ \sqrt{(a)^2 + (0.5)^2 + (0.5)^2} = 1 \]

### Step 4: Square both sides to eliminate the square root
Squaring both sides gives us:
\[ (a)^2 + (0.5)^2 + (0.5)^2 = 1 \]

### Step 5: Calculate the squares of the components
Calculating \( (0.5)^2 \):
\[ (0.5)^2 = 0.25 \]
Thus, we can rewrite the equation as:
\[ a^2 + 0.25 + 0.25 = 1 \]

### Step 6: Combine like terms
Combining the constants gives:
\[ a^2 + 0.5 = 1 \]

### Step 7: Isolate \( a^2 \)
Subtract \( 0.5 \) from both sides:
\[ a^2 = 1 - 0.5 \]
\[ a^2 = 0.5 \]

### Step 8: Solve for \( a \)
To find \( a \), take the square root of both sides:
\[ a = \sqrt{0.5} = \frac{1}{\sqrt{2}} \]

### Conclusion
Thus, the value of \( a \) is:
\[ a = \frac{1}{\sqrt{2}} \]

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