If a vector `vecr` has magnitude 14 and direction ratios 2, 3 and -6. Then, find the direction cosines and components of `vecr`, given that `vecr` makes an acute angle with X-axis.

Answer

To solve the problem, we need to find the direction cosines and components of the vector \(\vec{r}\) given its magnitude and direction ratios. Let's break down the solution step by step.

### Step 1: Understand the Given Information
We are given:
- Magnitude of the vector \(\vec{r} = 14\)
- Direction ratios of the vector \(\vec{r} = 2, 3, -6\)

### Step 2: Relate Direction Ratios to Components
The direction ratios can be expressed in terms of a parameter \(\lambda\):
- Let the components of the vector \(\vec{r}\) be \(a, b, c\).
- We can set:
\[ a = 2\lambda, \quad b = 3\lambda, \quad c = -6\lambda \]

### Step 3: Calculate the Magnitude of the Vector
The magnitude of the vector \(\vec{r}\) is given by:
\[ |\vec{r}| = \sqrt{a^2 + b^2 + c^2} \]
Substituting the expressions for \(a, b, c\):
\[ |\vec{r}| = \sqrt{(2\lambda)^2 + (3\lambda)^2 + (-6\lambda)^2} \]
Calculating each term:
\[ = \sqrt{4\lambda^2 + 9\lambda^2 + 36\lambda^2} = \sqrt{49\lambda^2} = 7|\lambda| \]
Since the magnitude is given as 14, we set:
\[ 7|\lambda| = 14 \]
Thus, we find:
\[ |\lambda| = 2 \]
Since \(\vec{r}\) makes an acute angle with the X-axis, we take \(\lambda = 2\).

### Step 4: Find the Components of the Vector
Now substituting \(\lambda = 2\) back into the expressions for \(a, b, c\):
\[ a = 2 \cdot 2 = 4, \quad b = 3 \cdot 2 = 6, \quad c = -6 \cdot 2 = -12 \]
Thus, the components of the vector \(\vec{r}\) are:
\[ \vec{r} = 4\hat{i} + 6\hat{j} - 12\hat{k} \]

### Step 5: Find the Direction Cosines
The direction cosines \(l, m, n\) are given by:
\[ l = \frac{a}{|\vec{r}|}, \quad m = \frac{b}{|\vec{r}|}, \quad n = \frac{c}{|\vec{r}|} \]
Substituting the values:
\[ l = \frac{4}{14} = \frac{2}{7}, \quad m = \frac{6}{14} = \frac{3}{7}, \quad n = \frac{-12}{14} = -\frac{6}{7} \]

### Final Result
The direction cosines are:
\[ \left( \frac{2}{7}, \frac{3}{7}, -\frac{6}{7} \right) \]
And the components of the vector \(\vec{r}\) are:
\[ \vec{r} = 4\hat{i} + 6\hat{j} - 12\hat{k} \]