Q. 83.7( 3 Votes )

Show that the vec

Answer :

Vectors parallel to the same plane, or lie on the same plane are called coplanar vectors

The three vectors are coplanar if one of them is expressible as a linear combination of the other two.


Given that





Let






Comparing the coefficients of , and , we get


1 = 2x + y …(1)


2 = x + y …(2)


3 = 3x + y …(3)


Solving equation (1) and (2), we get



x = –1


Substitute x = –1 in equation (2), we get


2 = x + y


2 = –1 + y


y = 2 + 1


y = 3


Put x = –1 and y = 3 in equation (3), we get


3 = 3x + y


3 = 3(–1) + 3


3 = –3 + 3


3 ≠ 0


L.H.S ≠ R.H.S


The value of x and y doesn’t satisfy equation (3).


Thus, , and are not coplanar.


Let be depicted as,


…(*)


Substitute the value of , , and .





Comparing the coefficients in , and , we get


2 = x + 2y + z …(1)


–1 = 2x + y + z …(2)


–3 = 3x + 3y + z …(3)


From equation (1),


2 = x + 2y + z


z = 2 – x – 2y …(4)


Putting the value of z from equation (4) in equations (2) & (3), we get


From equation (2),


–1 = 2x + y + z


–1 = 2x + y + (2 – x – 2y)


–1 = 2x + y + 2 – x – 2y


2x – x + y – 2y = –1 – 2


x – y = –3 …(5)


From equation (3),


–3 = 3x + 3y + z


–3 = 3x + 3y + (2 – x – 2y)


–3 = 3x + 3y + 2 – x – 2y


3x – x + 3y – 2y = –3 – 2


2x + y = –5 …(6)


Solving equation (5) and (6), we have



3x = –8



Substituting in equation (5), we get


x – y = –3




–8 – 3y = –3 × 3


–8 – 3y = –9


3y = 9 – 8


3y = 1



Now, substitute and in z = 2 – x – 2y, we get








z = 4


We have got , and z = 4.


Put these values in equation (*), we get



Thus, we have found the relation.


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