If `vec(a)=hat(i)+2hat(j)`, then `|vec(a)|` is :

Options

A. 3
B. `-1`
C. 5
D. `sqrt(5)`

Answer

To find the magnitude of the vector \(\vec{a} = \hat{i} + 2\hat{j}\), we will follow these steps:

### Step 1: Identify the components of the vector
The vector \(\vec{a}\) can be expressed in terms of its components:
- The coefficient of \(\hat{i}\) is 1.
- The coefficient of \(\hat{j}\) is 2.
- There is no \(\hat{k}\) component, so it is 0.

Thus, we can write:
\[ \vec{a} = 1\hat{i} + 2\hat{j} + 0\hat{k} \]

### Step 2: Use the formula for the magnitude of a vector
The magnitude \(|\vec{a}|\) of a vector \(\vec{a} = x\hat{i} + y\hat{j} + z\hat{k}\) is given by the formula:
\[ |\vec{a}| = \sqrt{x^2 + y^2 + z^2} \]

### Step 3: Substitute the components into the formula
Substituting the values of \(x\), \(y\), and \(z\):
- \(x = 1\)
- \(y = 2\)
- \(z = 0\)

We have:
\[ |\vec{a}| = \sqrt{(1)^2 + (2)^2 + (0)^2} \]

### Step 4: Calculate the squares
Calculating the squares:
\[ |\vec{a}| = \sqrt{1^2 + 2^2 + 0^2} = \sqrt{1 + 4 + 0} \]

### Step 5: Simplify the expression
Now, simplify the expression:
\[ |\vec{a}| = \sqrt{5} \]

### Final Answer
Thus, the magnitude of the vector \(\vec{a}\) is:
\[ |\vec{a}| = \sqrt{5} \]

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