If the projections of a line segment on the x, y and z-axes in 3-dimensional space are 2, 3 and 6 respectively, then the length of the line segment is :

We are given that the projections of a line segment on the x, y, and z-axes are 2, 3, and 6 respectively. In 3-dimensional space, the projection on an axis represents the absolute difference in the coordinates of the endpoints along that axis. So, for a line segment from point A(x1, y1, z1) to point B(x2, y2, z2), we have:

The projection on the x-axis is |x2 - x1| = 2.

The projection on the y-axis is |y2 - y1| = 3.

The projection on the z-axis is |z2 - z1| = 6.

The actual length of the line segment is the distance between points A and B, which is given by the distance formula:

$$ \text{Length} = \sqrt{(x2 - x1)^2 + (y2 - y1)^2 + (z2 - z1)^2} $$

Since we have the absolute values, we can substitute the given projections directly into the formula. Let:

$$ a = |x2 - x1| = 2 $$

$$ b = |y2 - y1| = 3 $$

$$ c = |z2 - z1| = 6 $$

Then the length L is:

$$ L = \sqrt{a^2 + b^2 + c^2} $$

Substituting the values:

$$ L = \sqrt{(2)^2 + (3)^2 + (6)^2} $$

Calculating the squares:

$$ (2)^2 = 4 $$

$$ (3)^2 = 9 $$

$$ (6)^2 = 36 $$

Adding these:

$$ 4 + 9 + 36 = 49 $$

So,

$$ L = \sqrt{49} = 7 $$

Therefore, the length of the line segment is 7.

Comparing with the options:

A. 12

B. 7

C. 9

D. 6

Hence, the correct answer is Option B.