Let the foot of the perpendicular from the point \(P(0, 1, 2)\) on the x-axis be \(Q(x, 0, 0)\).
Since \(Q\) lies on the x-axis, its y and z coordinates are 0.
The direction ratios of the line \(PQ\) are \((x - 0, 0 - 1, 0 - 2)\), which simplifies to \((x, -1, -2)\).
Since \(PQ\) is perpendicular to the x-axis, the direction ratios of \(PQ\) are perpendicular to the direction ratios of the x-axis, which are \((1, 0, 0)\).
The dot product of the direction ratios of \(PQ\) and the x-axis must be 0: \(x(1) + (-1)(0) + (-2)(0) = 0\).
This simplifies to \(x = 0\).
Therefore, the coordinates of the foot of the perpendicular are \((0, 0, 0)\).
Correct Answer: (0,0,0)
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Pedagogical Audit
Bloom's Analysis:
This is an APPLY question because it requires students to apply their understanding of perpendicularity and coordinate geometry to find the foot of the perpendicular.
Knowledge Dimension:PROCEDURAL
Justification:The question requires the student to follow a specific procedure to determine the coordinates of the foot of the perpendicular. This involves understanding the properties of the x-axis and perpendicular lines in 3D space.
Syllabus Audit:
In the context of CBSE Class 12, this is classified as KNOWLEDGE. The question directly assesses the student's understanding of coordinate geometry concepts as taught in the textbook.