Find the angle between the lines `(x+3)/(3)=(y+1)/(5)=(z+3)/(4)and(x+1)/(1)=(y-4)/(1)=(z-5)/(2)`.

Answer

To find the angle between the two lines given by the equations

\[ \frac{x+3}{3} = \frac{y+1}{5} = \frac{z+3}{4} \]

and

\[ \frac{x+1}{1} = \frac{y-4}{1} = \frac{z-5}{2}, \]

we can follow these steps:

### Step 1: Identify the direction ratios of the lines

For the first line, the direction ratios can be extracted from the coefficients of \(x\), \(y\), and \(z\):

- From \(\frac{x+3}{3}\), the coefficient is \(3\).
- From \(\frac{y+1}{5}\), the coefficient is \(5\).
- From \(\frac{z+3}{4}\), the coefficient is \(4\).

Thus, the direction ratios for the first line \(b_1\) are:

\[ b_1 = (3, 5, 4). \]

For the second line, the direction ratios are:

- From \(\frac{x+1}{1}\), the coefficient is \(1\).
- From \(\frac{y-4}{1}\), the coefficient is \(1\).
- From \(\frac{z-5}{2}\), the coefficient is \(2\).

Thus, the direction ratios for the second line \(b_2\) are:

\[ b_2 = (1, 1, 2). \]

### Step 2: Calculate the dot product of \(b_1\) and \(b_2\)

The dot product \(b_1 \cdot b_2\) is calculated as follows:

\[ b_1 \cdot b_2 = (3)(1) + (5)(1) + (4)(2) = 3 + 5 + 8 = 16. \]

### Step 3: Calculate the magnitudes of \(b_1\) and \(b_2\)

The magnitude of \(b_1\) is:

\[ |b_1| = \sqrt{3^2 + 5^2 + 4^2} = \sqrt{9 + 25 + 16} = \sqrt{50} = 5\sqrt{2}. \]

The magnitude of \(b_2\) is:

\[ |b_2| = \sqrt{1^2 + 1^2 + 2^2} = \sqrt{1 + 1 + 4} = \sqrt{6}. \]

### Step 4: Use the formula to find \(\cos \theta\)

The cosine of the angle \(\theta\) between the two lines can be found using the formula:

\[ \cos \theta = \frac{b_1 \cdot b_2}{|b_1| |b_2|}. \]

Substituting the values we calculated:

\[ \cos \theta = \frac{16}{(5\sqrt{2})(\sqrt{6})} = \frac{16}{5\sqrt{12}} = \frac{16}{5 \cdot 2\sqrt{3}} = \frac{8}{5\sqrt{3}}. \]

### Step 5: Conclusion

Thus, the angle \(\theta\) between the two lines is given by:

\[ \cos \theta = \frac{8}{5\sqrt{3}}. \]