Question

Equation of line through (0, 0, 1) and (1, 1, 0)

• A. \( x + y + z = 1 \)
• B. \( x + y + z = 0 \)
• C. \( x - y + z = 0 \)
• D. \( x + y - z = 0 \)

Hint:

When finding the equation of a line through two points in 3D, parametrize the line and then eliminate the parameter to get the desired equation.

Answer 1

The Correct Option is A

The equation of the line through two points \( P_1 = (0, 0, 1) \) and \( P_2 = (1, 1, 0) \) can be written as: \[ \frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1} \] Substituting the coordinates of \(P_1\) and \(P_2\): \[ \frac{x - 0}{1 - 0} = \frac{y - 0}{1 - 0} = \frac{z - 1}{0 - 1} \] This simplifies to: \[ x = y, \quad z = 1 - x \] Thus, the equation of the line is: \[ x + y + z = 1 \]