Question

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. $\frac{x+2}{3}=\frac{y}{13}=\frac{z-11}{-7}$

Hint:

Always ensure signs are reversed for the coordinates in the numerator $(x-x_1)$.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
We need to convert the vector equation of a line into its Cartesian form. The vector equation gives us a point on the line and the direction ratios of the line.
Step 2: Key Formula or Approach:
The vector equation of a line is \( \vec{r} = \vec{a} + \lambda\vec{b} \), where:
- \( \vec{r} = x\hat{i} + y\hat{j} + z\hat{k} \) is the position vector of any point on the line.
- \( \vec{a} = x_1\hat{i} + y_1\hat{j} + z_1\hat{k} \) is the position vector of a specific point \( (x_1, y_1, z_1) \) on the line.
- \( \vec{b} = l\hat{i} + m\hat{j} + n\hat{k} \) is the direction vector of the line, where \( (l, m, n) \) are the direction ratios.
The corresponding Cartesian equation is:
\[ \frac{x - x_1}{l} = \frac{y - y_1}{m} = \frac{z - z_1}{n} \] Step 3: Detailed Explanation:
The given vector equation is:
\[ \vec{r} = (2\hat{i} - 7\hat{j} + 11\hat{k}) + \lambda(3\hat{i} + 7\hat{j} - 13\hat{k}) \] By comparing this with the standard form \( \vec{r} = \vec{a} + \lambda\vec{b} \), we can identify:
The point on the line corresponds to \( \vec{a} \), so \( (x_1, y_1, z_1) = (2, -7, 11) \).
The direction vector is \( \vec{b} \), so the direction ratios are \( (l, m, n) = (3, 7, -13) \).
Now, substitute these values into the Cartesian formula:
\[ \frac{x - 2}{3} = \frac{y - (-7)}{7} = \frac{z - 11}{-13} \] Simplifying the y-term gives:
\[ \frac{x - 2}{3} = \frac{y + 7}{7} = \frac{z - 11}{-13} \] Step 4: Final Answer:
The Cartesian equation of the line is \( \frac{x-2}{3} = \frac{y+7}{7} = \frac{z-11}{-13} \).