If `vec(a)=2hat(i)+3hat(j)-hat(k)`, then `|vec(a)|` is :

Options

A. `sqrt(15)`
B. `sqrt(14)`
C. 14
D. 15

Answer

To find the magnitude of the vector \(\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}\), we will follow these steps:

### Step 1: Identify the components of the vector
The vector \(\vec{a}\) can be broken down into its components:
- \(x = 2\) (coefficient of \(\hat{i}\))
- \(y = 3\) (coefficient of \(\hat{j}\))
- \(z = -1\) (coefficient of \(\hat{k}\))

### 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 values into the formula
Substituting the values of \(x\), \(y\), and \(z\) into the formula:
\[ |\vec{a}| = \sqrt{(2)^2 + (3)^2 + (-1)^2} \]

### Step 4: Calculate the squares of the components
Calculating the squares:
- \(2^2 = 4\)
- \(3^2 = 9\)
- \((-1)^2 = 1\)

### Step 5: Sum the squares
Now add these values together:
\[ |\vec{a}| = \sqrt{4 + 9 + 1} = \sqrt{14} \]

### Step 6: Final result
Thus, the magnitude of the vector \(\vec{a}\) is:
\[ |\vec{a}| = \sqrt{14} \]