Direction Cosines And Direction Ratios Of A Line

Step 1: Recall the property of direction cosines.
If are the direction cosines of a line, then they must satisfy the fundamental identity This condition is necessary because direction cosines represent the cosines of angles that a line makes with the coordinate axes.
Step 2: Test the assertion.
The assertion claims that the direction cosines of a line can be Check the condition: But the required value must be Thus the condition for direction cosines is not satisfied.
Therefore the statement that a line can have direction cosines is false.
Step 3: Check the reason statement.
The reason states This occurs when This is a standard trigonometric result and is mathematically correct.
Thus the reason statement is true.
Step 4: Final conclusion.
Since the assertion is false but the reason is true, the correct option becomes Final Answer:

Step 1: Express the line in parametric form.
Given Let Then Thus So the parametric form becomes Step 2: Find direction ratios.
Direction ratios are coefficients of .
Thus direction ratios are Step 3: Convert to direction cosines.
Magnitude of vector Therefore direction cosines are Conclusion:
Hence the direction cosines are Final Answer: