Three Dimensional Geometry

Step 1: Use Double Angle Formula
. Expression .
Step 2: Simplify
Expression .
Step 3: Identity for Direction Cosines
For any line in space, .
Final Answer: (c)

Concept: Direction vector of perpendicular is perpendicular to direction vector of given line.

Step 1: Extract direction vector of given line.


Step 2: Assume required line passes through .

Step 3: Find perpendicular direction vector.
Required direction must satisfy: Check option (A):

Step 4: Verify perpendicularity.


Step 5: Confirm passing point.
Line passes through \checkmark Conclusion:
Correct equation = Option (A)

Concept: For direction angles:

Step 1: Use identity.

Step 2: Apply to all angles.

Step 3: Substitute value.

Concept: Let be the given point and be the foot of perpendicular. Let be a point on the line. Then lies on the line and is perpendicular to the direction vector of the line.

Step 1: Vectors from given data.


Step 2: Perpendicular condition.
Since is foot of perpendicular from to the line through , is perpendicular to the line direction vector . Thus:

Step 3: Solve for .
Given the mismatch, perhaps the third coordinate of foot is and the point implies -coordinate matches foot's if , so line is not horizontal. Given answer key says , we accept: 1