The magnitude of the vector `6 hat(i) - 2hat(j) + 3hat(k)` is a) 5 units b) 7 units c) 11 units d) 1 unit

Options

A. 5 units
B. 7 units
C. 11 units
D. 1 unit

Answer

To find the magnitude of the vector \( \mathbf{a} = 6\hat{i} - 2\hat{j} + 3\hat{k} \), we will follow these steps:

### Step 1: Identify the components of the vector
The vector \( \mathbf{a} \) has the following components:
- \( a_x = 6 \) (coefficient of \( \hat{i} \))
- \( a_y = -2 \) (coefficient of \( \hat{j} \))
- \( a_z = 3 \) (coefficient of \( \hat{k} \))

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

### Step 3: Substitute the components into the formula
Substituting the values of \( a_x \), \( a_y \), and \( a_z \):
\[ |\mathbf{a}| = \sqrt{6^2 + (-2)^2 + 3^2} \]

### Step 4: Calculate the squares of the components
Calculating each square:
- \( 6^2 = 36 \)
- \( (-2)^2 = 4 \)
- \( 3^2 = 9 \)

### Step 5: Sum the squares
Now, we sum these values:
\[ 36 + 4 + 9 = 49 \]

### Step 6: Take the square root
Finally, we take the square root of the sum:
\[ |\mathbf{a}| = \sqrt{49} = 7 \]

### Conclusion
The magnitude of the vector \( \mathbf{a} = 6\hat{i} - 2\hat{j} + 3\hat{k} \) is \( 7 \) units.

### Answer
The correct option is (b) 7 units.

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