Find the cartesian and vector equations of the line passing through \(A(1, 2, 3)\) and having direction ratios 2, 3, 7.

Answer: \(\bar r = (\hat i + 2\hat j + 3\hat k) + \lambda(2\hat i + 3\hat j + 7\hat k)\); \(\dfrac{x - 1}{2} = \dfrac{y - 2}{3} = \dfrac{z - 3}{7}\)

Position vector of \(A\): \(\bar a = \hat i + 2\hat j + 3\hat k\). A vector along the line: \(\bar b = 2\hat i + 3\hat j + 7\hat k\).

Vector equation: \(\bar r = \bar a + \lambda\bar b = (\hat i + 2\hat j + 3\hat k) + \lambda(2\hat i + 3\hat j + 7\hat k)\).

Cartesian equations: \(\dfrac{x - x_1}{a} = \dfrac{y - y_1}{b} = \dfrac{z - z_1}{c}\), i.e. \(\dfrac{x - 1}{2} = \dfrac{y - 2}{3} = \dfrac{z - 3}{7}\).