The direction-cosines of the vector `hati + 2 hatj + 3 hatk ` are `lt (1)/(sqrt(14)), (2)/(sqrt(14)),(3)/(sqrt(14)) gt`

Answer

To determine whether the statement about the direction cosines of the vector \( \hat{i} + 2\hat{j} + 3\hat{k} \) is true or false, we need to calculate the direction cosines of the given vector.

### Step-by-Step Solution:

1. **Identify the vector components**:
The given vector is \( \mathbf{v} = \hat{i} + 2\hat{j} + 3\hat{k} \).
Here, the components are:
- \( a = 1 \) (coefficient of \( \hat{i} \))
- \( b = 2 \) (coefficient of \( \hat{j} \))
- \( c = 3 \) (coefficient of \( \hat{k} \))

2. **Calculate the magnitude of the vector**:
The magnitude \( |\mathbf{v}| \) of the vector can be calculated using the formula:
\[ |\mathbf{v}| = \sqrt{a^2 + b^2 + c^2} \]
Substituting the values:
\[ |\mathbf{v}| = \sqrt{1^2 + 2^2 + 3^2} = \sqrt{1 + 4 + 9} = \sqrt{14} \]

3. **Calculate the direction cosines**:
The direction cosines \( l, m, n \) are given by the formulas:
\[ l = \frac{a}{|\mathbf{v}|}, \quad m = \frac{b}{|\mathbf{v}|}, \quad n = \frac{c}{|\mathbf{v}|} \]
Substituting the values we found:
\[ l = \frac{1}{\sqrt{14}}, \quad m = \frac{2}{\sqrt{14}}, \quad n = \frac{3}{\sqrt{14}} \]

4. **Conclusion**:
The calculated direction cosines are:
\[ \left( \frac{1}{\sqrt{14}}, \frac{2}{\sqrt{14}}, \frac{3}{\sqrt{14}} \right) \]
This matches the statement given in the question. Therefore, the statement is **true**.