(i) Find the distance from (1,2,3 ) to the plane 2x + 3y - z + 2 = 0 .
Find the length of perpendicular drawn from the origin to the plane 2x - 3y + 6z + 21 = 0.

Answer

To solve the given problems, we will use the formula for the distance from a point to a plane in three-dimensional geometry.

### Problem (i): Find the distance from the point (1, 2, 3) to the plane \(2x + 3y - z + 2 = 0\).

1. **Identify the coefficients from the plane equation**:
The equation of the plane is given as \(2x + 3y - z + 2 = 0\). Here, we can identify:
- \(a = 2\)
- \(b = 3\)
- \(c = -1\)
- \(d = 2\)

2. **Use the distance formula**:
The formula for the distance \(D\) from a point \((x_1, y_1, z_1)\) to the plane \(ax + by + cz + d = 0\) is given by:
\[ D = \frac{|ax_1 + by_1 + cz_1 + d|}{\sqrt{a^2 + b^2 + c^2}} \]
Substituting the point \((1, 2, 3)\) into the formula:
- \(x_1 = 1\)
- \(y_1 = 2\)
- \(z_1 = 3\)

Now, calculate:
\[ D = \frac{|2(1) + 3(2) - 1(3) + 2|}{\sqrt{2^2 + 3^2 + (-1)^2}} \]

3. **Calculate the numerator**:
\[ 2(1) + 3(2) - 1(3) + 2 = 2 + 6 - 3 + 2 = 7 \]
Thus, the absolute value is \(|7| = 7\).

4. **Calculate the denominator**:
\[ \sqrt{2^2 + 3^2 + (-1)^2} = \sqrt{4 + 9 + 1} = \sqrt{14} \]

5. **Combine the results**:
\[ D = \frac{7}{\sqrt{14}} = \frac{7\sqrt{14}}{14} = \frac{\sqrt{14}}{2} \]

### Problem (ii): Find the length of the perpendicular drawn from the origin to the plane \(2x - 3y + 6z + 21 = 0\).

1. **Identify the coefficients from the plane equation**:
The equation of the plane is \(2x - 3y + 6z + 21 = 0\). Here, we can identify:
- \(a = 2\)
- \(b = -3\)
- \(c = 6\)
- \(d = 21\)

2. **Use the distance formula**:
The distance \(D\) from the origin \((0, 0, 0)\) to the plane is given by:
\[ D = \frac{|a(0) + b(0) + c(0) + d|}{\sqrt{a^2 + b^2 + c^2}} \]

3. **Calculate the numerator**:
\[ D = \frac{|0 + 0 + 0 + 21|}{\sqrt{2^2 + (-3)^2 + 6^2}} = \frac{21}{\sqrt{4 + 9 + 36}} = \frac{21}{\sqrt{49}} = \frac{21}{7} = 3 \]

### Final Answers:
1. The distance from the point (1, 2, 3) to the plane \(2x + 3y - z + 2 = 0\) is \(\frac{\sqrt{14}}{2}\) units.
2. The length of the perpendicular drawn from the origin to the plane \(2x - 3y + 6z + 21 = 0\) is \(3\) units.