CBSE Class 12 Math 2009 Solved Paper - Question 16 | ExamSiri

CBSE Class 12 Math 2009 Solved Paper

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Question : 16 of 29

Marks: +1, -0

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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