The equation of a line are 5x-3 =15y+7=3-10z write the direction cosines of the line

Answer

To find the direction cosines of the line given by the equation \( 5x - 3 = 15y + 7 = 3 - 10z \), we can follow these steps:

### Step 1: Rewrite the equation in parametric form
We start by rewriting the equation in a more manageable form. We can set:
\[ 5x - 3 = k, \quad 15y + 7 = k, \quad 3 - 10z = k \]
where \( k \) is a parameter.

From these equations, we can express \( x \), \( y \), and \( z \) in terms of \( k \):

1. From \( 5x - 3 = k \):
\[ 5x = k + 3 \implies x = \frac{k + 3}{5} \]

2. From \( 15y + 7 = k \):
\[ 15y = k - 7 \implies y = \frac{k - 7}{15} \]

3. From \( 3 - 10z = k \):
\[ 10z = 3 - k \implies z = \frac{3 - k}{10} \]

### Step 2: Identify the direction ratios
The direction ratios of the line can be obtained from the coefficients of \( k \) in the parametric equations. Thus, we have:
\[ x = \frac{1}{5}k + \frac{3}{5}, \quad y = \frac{1}{15}k - \frac{7}{15}, \quad z = -\frac{1}{10}k + \frac{3}{10} \]

The direction ratios \( (a, b, c) \) are:
\[ a = \frac{1}{5}, \quad b = \frac{1}{15}, \quad c = -\frac{1}{10} \]

### Step 3: Calculate the direction cosines
The direction cosines \( l, m, n \) are given by:
\[ l = \frac{a}{\sqrt{a^2 + b^2 + c^2}}, \quad m = \frac{b}{\sqrt{a^2 + b^2 + c^2}}, \quad n = \frac{c}{\sqrt{a^2 + b^2 + c^2}} \]

First, we calculate \( a^2 + b^2 + c^2 \):
\[ a^2 = \left(\frac{1}{5}\right)^2 = \frac{1}{25}, \quad b^2 = \left(\frac{1}{15}\right)^2 = \frac{1}{225}, \quad c^2 = \left(-\frac{1}{10}\right)^2 = \frac{1}{100} \]

Now, we find a common denominator to add these fractions:
\[ a^2 + b^2 + c^2 = \frac{9}{225} + \frac{1}{225} + \frac{25}{225} = \frac{35}{225} = \frac{7}{45} \]

Now, we find the square root:
\[ \sqrt{a^2 + b^2 + c^2} = \sqrt{\frac{7}{45}} = \frac{\sqrt{7}}{3\sqrt{5}} = \frac{\sqrt{35}}{15} \]

Now we can calculate the direction cosines:
\[ l = \frac{\frac{1}{5}}{\frac{\sqrt{35}}{15}} = \frac{3}{\sqrt{35}}, \quad m = \frac{\frac{1}{15}}{\frac{\sqrt{35}}{15}} = \frac{1}{\sqrt{35}}, \quad n = \frac{-\frac{1}{10}}{\frac{\sqrt{35}}{15}} = -\frac{3}{\sqrt{35}} \]

### Final Result
The direction cosines of the line are:
\[ l = \frac{3}{\sqrt{35}}, \quad m = \frac{1}{\sqrt{35}}, \quad n = -\frac{3}{\sqrt{35}} \]