Find the vector equation of a line passing through the point (2,3,2) and parallel to the line :
`vec(r) = (-2 hat(i) + 3 hat(j) ) + lambda (2 hat(i) - 3 hat(j) + 6 hat(k))` .
Also, find the distnace between these two lines.

Answer

To find the vector equation of a line passing through the point (2, 3, 2) and parallel to the given line, we will follow these steps:

### Step 1: Identify the direction vector of the given line

The given line is represented as:
\[ \vec{r} = (-2 \hat{i} + 3 \hat{j}) + \lambda (2 \hat{i} - 3 \hat{j} + 6 \hat{k}) \]
From this equation, we can see that the direction vector of the line is:
\[ \vec{d} = 2 \hat{i} - 3 \hat{j} + 6 \hat{k} \]

### Step 2: Write the vector equation of the new line

The vector equation of a line can be written in the form:
\[ \vec{r} = \vec{a} + \mu \vec{d} \]
where \(\vec{a}\) is a point on the line and \(\vec{d}\) is the direction vector.

Here, the point \(\vec{a}\) is (2, 3, 2), which can be expressed as:
\[ \vec{a} = 2 \hat{i} + 3 \hat{j} + 2 \hat{k} \]

Now substituting \(\vec{a}\) and \(\vec{d}\) into the equation, we get:
\[ \vec{r} = (2 \hat{i} + 3 \hat{j} + 2 \hat{k}) + \mu (2 \hat{i} - 3 \hat{j} + 6 \hat{k}) \]

### Step 3: Simplify the equation

We can simplify the equation:
\[ \vec{r} = (2 + 2\mu) \hat{i} + (3 - 3\mu) \hat{j} + (2 + 6\mu) \hat{k} \]

### Step 4: Find the distance between the two lines

To find the distance between two parallel lines, we can use the formula:
\[ \text{Distance} = \frac{|(\vec{b} - \vec{a}) \cdot (\vec{d} \times \vec{d'})|}{|\vec{d} \times \vec{d'}|} \]
where \(\vec{b}\) is a point on the first line, \(\vec{a}\) is a point on the second line, and \(\vec{d}\) and \(\vec{d'}\) are the direction vectors of the two lines.

Here, we can take:
- \(\vec{b} = -2 \hat{i} + 3 \hat{j}\) (from the first line)
- \(\vec{a} = 2 \hat{i} + 3 \hat{j} + 2 \hat{k}\)

Now, we calculate \(\vec{b} - \vec{a}\):
\[ \vec{b} - \vec{a} = (-2 \hat{i} + 3 \hat{j}) - (2 \hat{i} + 3 \hat{j} + 2 \hat{k}) = -4 \hat{i} - 2 \hat{k} \]

Next, we need to compute \(\vec{d} \times \vec{d'}\). Since both lines are parallel, \(\vec{d} = \vec{d'}\), thus \(\vec{d} \times \vec{d'} = \vec{0}\).

Since the direction vectors are the same, we can directly find the distance using the formula for the distance between a point and a line.

### Step 5: Calculate the distance

The distance from point \((2, 3, 2)\) to the line can be calculated using the formula:
\[ \text{Distance} = \frac{|(-4 \hat{i} - 2 \hat{k}) \cdot (2 \hat{i} - 3 \hat{j} + 6 \hat{k})|}{|2 \hat{i} - 3 \hat{j} + 6 \hat{k}|} \]

Calculating the numerator:
\[ (-4 \hat{i} - 2 \hat{k}) \cdot (2 \hat{i} - 3 \hat{j} + 6 \hat{k}) = -4 \cdot 2 + 0 + (-2) \cdot 6 = -8 - 12 = -20 \]

Calculating the denominator:
\[ |2 \hat{i} - 3 \hat{j} + 6 \hat{k}| = \sqrt{2^2 + (-3)^2 + 6^2} = \sqrt{4 + 9 + 36} = \sqrt{49} = 7 \]

Thus, the distance is:
\[ \text{Distance} = \frac{|-20|}{7} = \frac{20}{7} \]

### Final Answer

The vector equation of the line is:
\[ \vec{r} = (2 + 2\mu) \hat{i} + (3 - 3\mu) \hat{j} + (2 + 6\mu) \hat{k} \]
And the distance between the two lines is:
\[ \frac{20}{7} \]