Q673: The equation of a line parallel to the vector \(3\hat{i}+\ha...

Class CBSE Class 12 Mathematics Three Dimensional Geometry Q #673

COMPETENCY BASED

APPLY

1 Marks 2025 AISSCE(Board Exam) MCQ SINGLE

The equation of a line parallel to the vector $3\hat{i}+\hat{j}+2\hat{k}$ and passing through the point $(4, -3, 7)$ is:

(A) $x=4t+3, y=-3t+1, z=7t+2$

(B) $x=3t+4, y=t+3, z=2t+7$

(D) $x=3t+4, y=-t+3, z=2t+7$

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Correct Answer: C

Explanation

The vector equation of a line passing through point $\vec{a}$ and parallel to vector $\vec{b}$ is $\vec{r} = \vec{a} + t\vec{b}$.
Here, $\vec{a} = (4, -3, 7)$ and $\vec{b} = (3, 1, 2)$.
So, $(x, y, z) = (4, -3, 7) + t(3, 1, 2) = (4+3t, -3+t, 7+2t)$.
Thus, $x = 3t+4, y = t-3, z = 2t+7$.
This matches Option C.

AI Tutor Explanation

Step-by-Step Solution

The equation of a line passing through a point $\vec{a}$ and parallel to a vector $\vec{b}$ is given by $\vec{r} = \vec{a} + t\vec{b}$, where $t$ is a scalar.

Here, the point is $(4, -3, 7)$, so $\vec{a} = 4\hat{i} - 3\hat{j} + 7\hat{k}$.

The vector parallel to the line is $3\hat{i} + \hat{j} + 2\hat{k}$, so $\vec{b} = 3\hat{i} + \hat{j} + 2\hat{k}$.

Therefore, the equation of the line is $\vec{r} = (4\hat{i} - 3\hat{j} + 7\hat{k}) + t(3\hat{i} + \hat{j} + 2\hat{k})$.

In component form, this is:

$x = 4 + 3t$

$y = -3 + t$

$z = 7 + 2t$

Rearranging, we get:

$x = 3t + 4$

$y = t - 3$

$z = 2t + 7$

Correct Answer: x=3t+4, y=t-3, z=2t+7

AI Suggestion: Option C

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Pedagogical Audit

Bloom's Analysis: This is an APPLY question because it requires students to apply their understanding of vector equations of lines to find the correct parametric equation given a point and a parallel vector.

Knowledge Dimension: PROCEDURAL

Justification: The question requires the student to apply a specific procedure (forming the equation of a line given a point and a parallel vector) to arrive at the solution.

Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. The question assesses the student's ability to apply the concepts of vectors and 3D geometry to solve a specific problem, rather than simply recalling a definition or theorem.

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