Question

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

• A. \(\frac{x-3}{2}=\frac{y-2}{-1},\ z=1\)
• B. \(\frac{x-3}{2}=\frac{y-2}{-1},\ z=-1\)
• C. \(\frac{x-3}{2}=\frac{y-2}{1},\ z=-1\)
• D. \(\frac{x-3}{2}=\frac{y-2}{1},\ z=-1\)

Hint:

Find the direction ratios from the two points, and note that the z-coordinates are equal.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept
The line through \(A(x_1,y_1,z_1)\) and \(B(x_2,y_2,z_2)\) has direction ratios \((x_2-x_1,\ y_2-y_1,\ z_2-z_1)\).

Step 2: Direction Ratios
\(B-A=(1-3,\ 3-2,\ 1-1)=(-2,\ 1,\ 0)\).

Step 3: Write the Equation
\(\dfrac{x-3}{-2}=\dfrac{y-2}{1}\), and \(z=1\) because the z-direction ratio is 0.

Step 4: Match the Options
Multiplying numerator and denominator by \(-1\): \(\dfrac{x-3}{2}=\dfrac{y-2}{-1},\ z=1\).

Final Answer:
The line is \(\dfrac{x-3}{2}=\dfrac{y-2}{-1},\ z=1\). \[ \boxed{\text{(A) }\dfrac{x-3}{2}=\dfrac{y-2}{-1},\ z=1} \]