If distances of `(-1, 2, 3)` from `x, y, z` axis are `d_1,d_2, d_3` respectively and the distances from `xy, yz, zx` planes are `d_4, d_5, d_6` then the value of `sum_(i=1)^6 d_i` , is ___

Answer

To solve the problem, we need to find the distances \( d_1, d_2, d_3, d_4, d_5, \) and \( d_6 \) for the point \((-1, 2, 3)\) from the axes and planes, and then calculate the sum of these distances.

### Step 1: Calculate \( d_1 \) (Distance from the x-axis)
The distance from a point \((x, y, z)\) to the x-axis is given by the formula:
\[ d_1 = \sqrt{y^2 + z^2} \]
For the point \((-1, 2, 3)\):
\[ d_1 = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \]

### Step 2: Calculate \( d_2 \) (Distance from the y-axis)
The distance from a point \((x, y, z)\) to the y-axis is given by:
\[ d_2 = \sqrt{x^2 + z^2} \]
For the point \((-1, 2, 3)\):
\[ d_2 = \sqrt{(-1)^2 + 3^2} = \sqrt{1 + 9} = \sqrt{10} \]

### Step 3: Calculate \( d_3 \) (Distance from the z-axis)
The distance from a point \((x, y, z)\) to the z-axis is given by:
\[ d_3 = \sqrt{x^2 + y^2} \]
For the point \((-1, 2, 3)\):
\[ d_3 = \sqrt{(-1)^2 + 2^2} = \sqrt{1 + 4} = \sqrt{5} \]

### Step 4: Calculate \( d_4 \) (Distance from the xy-plane)
The distance from a point \((x, y, z)\) to the xy-plane is simply the absolute value of the z-coordinate:
\[ d_4 = |z| = |3| = 3 \]

### Step 5: Calculate \( d_5 \) (Distance from the yz-plane)
The distance from a point \((x, y, z)\) to the yz-plane is simply the absolute value of the x-coordinate:
\[ d_5 = |x| = |-1| = 1 \]

### Step 6: Calculate \( d_6 \) (Distance from the zx-plane)
The distance from a point \((x, y, z)\) to the zx-plane is simply the absolute value of the y-coordinate:
\[ d_6 = |y| = |2| = 2 \]

### Step 7: Calculate the sum of all distances
Now we can sum all the distances:
\[ \text{Sum} = d_1 + d_2 + d_3 + d_4 + d_5 + d_6 \]
Substituting the values we calculated:
\[ \text{Sum} = \sqrt{13} + \sqrt{10} + \sqrt{5} + 3 + 1 + 2 \]
\[ \text{Sum} = \sqrt{13} + \sqrt{10} + \sqrt{5} + 6 \]

Thus, the value of \( \sum_{i=1}^{6} d_i \) is:
\[ \sqrt{13} + \sqrt{10} + \sqrt{5} + 6 \]