Question

The Cartesian equation of a line is $3x + 1 = 6y - 2 = 1 - z$, then its vector equation is

• A. $\bar{r} = \left(-\frac{1}{3}\hat{i} + \frac{1}{3}\hat{j} + \hat{k}\right) + \lambda(2\hat{i} - \hat{j} - 6\hat{k})$
• B. $\bar{r} = (-\hat{i} + 2\hat{j} - \hat{k}) + \lambda(3\hat{i} + 6\hat{j} - \hat{k})$
• C. $\bar{r} = \left(-\frac{1}{3}\hat{i} + \frac{1}{3}\hat{j} + \hat{k}\right) + \lambda(2\hat{i} - \hat{j} + 6\hat{k})$
• D. $\bar{r} = \left(-\frac{1}{3}\hat{i} + \frac{1}{3}\hat{j} + \hat{k}\right) + \lambda(2\hat{i} + \hat{j} - 6\hat{k})$

Hint:

You can instantly eliminate options by checking the sign of the components. Notice that the $z$-term in the original equation is $1 - z = -(z - 1)$, meaning the $z$-direction component must carry a negative sign when standardizing. This leaves only options (A) and (D). Checking the $y$-coefficient isolation then gives the final result instantly.

Answer 1

The Correct Option is D

Step 1: Understanding the Question:
Convert the Cartesian line equation 3x + 1 = 6y - 2 = 1 - z into its vector form.

Step 2: Key Formula or Approach:
Rewrite in symmetric form (x - x₁)/a = (y - y₁)/b = (z - z₁)/c to extract a point (x₁, y₁, z₁) and direction ratios (a, b, c). The vector equation is r̄ = ā + λb̄.

Step 3: Detailed Explanation:
Factor coefficients: 3(x + 1/3) = 6(y - 1/3) = -1(z - 1). Dividing to get standard denominators: (x + 1/3)/(1/3) = (y - 1/3)/(1/6) = (z - 1)/(-1). Multiplying all denominators by 6 for integer ratios: (x + 1/3)/2 = (y - 1/3)/1 = (z - 1)/(-6). Thus point is (-1/3, 1/3, 1) and direction is (2, 1, -6). Vector equation: r̄ = (-1/3 î + 1/3 ĵ + k̂) + λ(2î + ĵ - 6k̂).

Step 4: Final Answer:
The vector equation matches option (D).