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The Cartesian equation of the line \(\vec{r} = (2\hat{i} - 7\hat{j} + 11\hat{k})\) \(+ \lambda(3\hat{i} + 7\hat{j} - 13\hat{k})\) is

  • A.\(\frac{x - 2}{3} = \frac{y + 7}{7} = \frac{z - 11}{-13}\)
  • B.\(\frac{x - 2}{3} = \frac{y - 7}{7} = \frac{z - 11}{13}\)
  • C.\(\frac{x + 2}{3} = \frac{y - 7}{7} = \frac{z + 11}{-13}\)
  • D.\(\frac{x + 2}{3} = \frac{y + 7}{7} = \frac{z - 11}{-13}\)
  • E.\(\frac{x + 2}{3} = \frac{y}{13} = \frac{z - 11}{-7}\)

Answer: A

The Cartesian form is \(\frac{x - 2}{3} = \frac{y + 7}{7} = \frac{z - 11}{-13}\).

A line through \((x_{1}, y_{1}, z_{1})\) with direction ratios a, b, c is \(\frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c}\). Here the point is (2, -7, 11) and the direction ratios are 3, 7, -13.

The y-term is \(y - (-7) = y + 7\). Options writing x + 2 or y - 7 reverse the signs of the point's coordinates, the option with 13 in the z-denominator loses the minus sign of the direction ratio, and the last option uses neither the point nor the direction.

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