The Cartesian equation of the line passing through the point (1, -3, 2) and parallel to the line: \(\vec{r}=(2+\lambda)\hat{i}+\lambda\hat{j}+(2\lambda-1)\hat{k}\) is:
Find the direction vector of the given line:
The given line is \(\vec{r}=(2+\lambda)\hat{i}+\lambda\hat{j}+(2\lambda-1)\hat{k} = (2\hat{i} - \hat{k}) + \lambda(\hat{i} + \hat{j} + 2\hat{k})\).
The direction vector of this line is \(\vec{b} = \hat{i} + \hat{j} + 2\hat{k}\). Therefore, the direction ratios are 1, 1, and 2.
Since the required line is parallel, it has the same direction ratios:
The required line is parallel to the given line, so its direction ratios are also 1, 1, and 2.
Use the point (1, -3, 2) and the direction ratios to form the Cartesian equation:
The Cartesian equation of a line passing through a point \((x_1, y_1, z_1)\) and having direction ratios a, b, c is given by:
\(\frac{x - x_1}{a} = \frac{y - y_1}{b} = \frac{z - z_1}{c}\)
In this case, \((x_1, y_1, z_1) = (1, -3, 2)\) and the direction ratios are a = 1, b = 1, c = 2.
Therefore, the equation of the line is:
\(\frac{x - 1}{1} = \frac{y - (-3)}{1} = \frac{z - 2}{2}\)
\(\frac{x - 1}{1} = \frac{y + 3}{1} = \frac{z - 2}{2}\)
Match the equation with the given options:
The equation \(\frac{x - 1}{1} = \frac{y + 3}{1} = \frac{z - 2}{2}\) matches option (D).
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Pedagogical Audit
Bloom's Analysis:
This is an APPLY question because it requires students to apply their knowledge of vector and Cartesian equations of lines to find the equation of a line given a point and a parallel vector.
Knowledge Dimension:PROCEDURAL
Justification:The question requires the student to follow a specific procedure to convert the vector equation to cartesian form and then use the point and direction ratios to find the required equation.
Syllabus Audit:
In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly tests the student's understanding of the concepts and formulas related to lines in 3D geometry as covered in the textbook.