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CBSE Class 12 Math 2009 Solved Paper

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Question : 16
Total: 29
If
→
a
×
→
b
 =
→
c
×
→
d
 and
→
a
×
→
c
 =
→
b
×
→
d
 , show that
→
a
−
→
d
 is parallel to
→
b
−
→
c
 , where
→
a
 ≠
→
d
 and
→
b
 ≠
→
c
Solution:  
Given :
→
a
×
→
b
 =
→
c
×
→
d
 and
→
a
×
→
c
 =
→
b
×
→
d
To show
→
a
−
→
d
 is parallel to
→
b
−
→
c
i.e (
→
a
−
→
d
) × (
→
b
−
→
c
) = 0
Consider (
→
a
−
→
d
) × (
→
b
−
→
c
) =
→
a
×(
→
b
−
→
c
)−
→
d
×(
→
b
−
→
c
)
=
→
a
×
→
b
 -
→
a
×
→
c
 -
→
d
×
→
b
 +
→
d
×
→
c
→
c
×
→
d
 -
→
b
×
→
d
 -
→
d
×
→
b
 +
→
d
×
→
c

[Since
→
a
×
→
b
 =
→
c
×
→
d
 and
→
a
×
→
c
 =
→
b
×
→
d
]
→
c
×
→
d
 -
→
b
×
→
d
 +
→
b
×
→
d
 -
→
c
×
→
d
[Since
→
d
×
→
c
 = −
→
c
×
→
d
 and
→
d
×
→
b
 = -
→
b
×
→
d
]
= 0
Therefore
→
a
−
→
d
 is parallel to
→
b
−
→
c
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