The projections of a vector on the three coordinate axis are $$6, -3, 2$$ respectively. The direction cosines of the vector are

1. Identify the Vector Components

The projections of a vector on the coordinate axes represent its Cartesian scalar components.

Let the vector be given as $$\vec{A}$$:

$$\vec{A} = 6\hat{i} - 3\hat{j} + 2\hat{k}$$

2. Calculate the Magnitude of the Vector

Find the total length of the vector using the three-dimensional distance formula:

$$|\vec{A}| = \sqrt{(6)^2 + (-3)^2 + (2)^2}$$

$$|\vec{A}| = \sqrt{36 + 9 + 4}$$

$$|\vec{A}| = \sqrt{49} = 7$$

3. Determine the Direction Cosines

The direction cosines $$(l, m, n)$$ are found by dividing each component by the vector magnitude:

$$l = \frac{6}{7}$$

$$m = -\frac{3}{7}$$

$$n = \frac{2}{7}$$

Final Answer

The direction cosines of the vector are $$\frac{6}{7}, -\frac{3}{7}, \frac{2}{7}$$.