To find the direction cosines of a line joining the points $(1, 2, 3)$ and $(2, 3, 2)$, follow these steps:

1. Calculate the direction ratios

The direction ratios of the line are given by the differences in the coordinates of the two points: $l = x_2 - x_1 = 2 - 1 = 1$ $m = y_2 - y_1 = 3 - 2 = 1$ $n = z_2 - z_1 = 2 - 3 = -1$ So, the direction ratios are $(1, 1, -1)$.

2. Calculate the magnitude of the direction ratios

The magnitude of the direction ratios is: $\sqrt{l^2 + m^2 + n^2} = \sqrt{1^2 + 1^2 + (-1)^2} = \sqrt{1 + 1 + 1} = \sqrt{3}$

3. Find the direction cosines

The direction cosines are the direction ratios divided by their magnitude: $\cos \alpha = \frac{l}{\sqrt{l^2 + m^2 + n^2}} = \frac{1}{\sqrt{3}}$ $\cos \beta = \frac{m}{\sqrt{l^2 + m^2 + n^2}} = \frac{1}{\sqrt{3}}$ $\cos \gamma = \frac{n}{\sqrt{l^2 + m^2 + n^2}} = \frac{-1}{\sqrt{3}}$

Thus, the direction cosines are: $\left(\frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{-1}{\sqrt{3}}\right)$

Would you like any further clarification or more details on the steps?

Related Questions:

1. How do you calculate the direction ratios if the points are in 2D?
2. What are direction cosines and how are they used in geometry?
3. Can you explain the relationship between direction cosines and direction angles?
4. How would the process change if the points were in 4D space?
5. What happens to the direction cosines if both points are identical?

Tip:

When solving 3D geometry problems, it's helpful to visualize the direction ratios as components of a vector that defines the orientation of the line.