To analyze these statements, we first determine the direction vectors for the lines joining these points.
Step 1: Find the direction vectors
The direction vector for a line passing through (x1,y1,z1) and (x2,y2,z2) is given by v=(x2−x1)i^+(y2−y1)j^+(z2−z1)k^.
Vector PQ: (4−(−2))i^+(−1−(−3))j^+(5−5)k^=6i^+2j^+0k^
Vector RS: (2−6)i^+(−6−(−4))j^+(10−8)k^=−4i^−2j^+2k^
Vector PR: (6−(−2))i^+(−4−(−3))j^+(8−5)k^=8i^−1j^+3k^
Vector QS: (2−4)i^+(−6−(−1))j^+(10−5)k^=−2i^−5j^+5k^
Step 2: Evaluate Statement I (Parallelism)
Two lines are parallel if their direction vectors are proportional.
PQ=6i^+2j^+0k^ and RS=−4i^−2j^+2k^.
Since −46=−22=20, the vectors are not proportional.
Statement I is incorrect.
Step 3: Evaluate Statement II (Perpendicularity)
Two lines are perpendicular if the dot product of their direction vectors is zero.
PR⋅QS=(8)(−2)+(−1)(−5)+(3)(5)
=−16+5+15
=−16+20
=4
Since the dot product is not zero (4=0), the lines are not perpendicular.
Statement II is incorrect.
Conclusion: Neither statement is correct.
Correct Option: (d) Neither I nor II
