Question

The Cartesian equation of the line passing through the points A(2, 2, 1) and B(1, 3, 0) is

• A. $\frac{x+2}{1} = \frac{y+2}{-1} = \frac{z+1}{-1}$
• B. $\frac{x-2}{-1} = \frac{y-2}{1} = \frac{z-1}{-1}$
• C. $\frac{x+2}{-1} = \frac{y+2}{1} = \frac{z+1}{-1}$
• D. None of these

Hint:

Always double-check your direction ratios by subtracting the coordinates in the same order. Using point B as the reference point instead of A would give an equivalent equation, but checking the options reveals that point A was used in the numerators.

Answer 1

The Correct Option is B

Step 1: Note the two points.
The line goes through $A(2, 2, 1)$ and $B(1, 3, 0)$.

Step 2: Find the direction.
Subtract the coordinates: direction $= B - A = (1-2,\ 3-2,\ 0-1) = (-1, 1, -1)$.

Step 3: Recall the Cartesian form.
A line through point $(x_1, y_1, z_1)$ with direction $(a, b, c)$ is
\[ \frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c} \]
Step 4: Use point $A$ and the direction.
\[ \frac{x - 2}{-1} = \frac{y - 2}{1} = \frac{z - 1}{-1} \]
Step 5: Compare with the options.
This matches the option that uses point $A(2,2,1)$ with direction numbers $-1, 1, -1$.

Step 6: Conclusion.
The other options use wrong signs in the numerators, so they are not correct. \[ \boxed{\frac{x-2}{-1} = \frac{y-2}{1} = \frac{z-1}{-1} \text{ (Option 2)}} \]