The D.C.'s of the vector `hat(i)+2hat(j)+3hat(k)` are :

Options

A. `(1)/(sqrt(6)),(2)/(sqrt(6)),(3)/(sqrt(6))`
B. `(1)/(sqrt(14)),(2)/(sqrt(14)),(3)/(sqrt(14))`
C. `1,2,3`
D. None of these

Answer

To find the direction cosines of the vector \(\hat{i} + 2\hat{j} + 3\hat{k}\), we can follow these steps:

### Step 1: Identify the vector
The given vector is:
\[ \mathbf{A} = \hat{i} + 2\hat{j} + 3\hat{k} \]

### Step 2: Calculate the magnitude of the vector
The magnitude of the vector \(\mathbf{A}\) is calculated using the formula:
\[ |\mathbf{A}| = \sqrt{a^2 + b^2 + c^2} \]
where \(a\), \(b\), and \(c\) are the coefficients of \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) respectively. For our vector:
- \(a = 1\)
- \(b = 2\)
- \(c = 3\)

Thus, the magnitude is:
\[ |\mathbf{A}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]

### Step 3: Find the direction cosines
The direction cosines \(l\), \(m\), and \(n\) are given by the formulas:
\[ l = \frac{a}{|\mathbf{A}|}, \quad m = \frac{b}{|\mathbf{A}|}, \quad n = \frac{c}{|\mathbf{A}|} \]

Substituting the values we have:
\[ l = \frac{1}{\sqrt{14}}, \quad m = \frac{2}{\sqrt{14}}, \quad n = \frac{3}{\sqrt{14}} \]

### Step 4: Write the final result
Thus, the direction cosines of the vector \(\hat{i} + 2\hat{j} + 3\hat{k}\) are:
\[ \left( \frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}} \right) \]