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If A = {0, 1, 2, 3, 5,6}, B = {1,3, 5, 7...

If A = {0, 1, 2, 3, 5,6}, B = {1,3, 5, 7, 9} and C = {0, 5, 10, 20,40}, find

A

1) A U B

B

2) A U C

C

3) B U C

D

4) A ∩ B

Text Solution

Verified by Experts

To show that P,Q,R,S are coplanar, we will show that PQ,PR,PS are coplanar
`PQ=-3b-2c`
`PR=2a-b+c`
`PS=4a+b+4c`
Let `PQ=xPR+yPS`
`implies-3b-2c=x(2a-b+c)+y(4a+b+4c)`
`implies-3b-2c=(2x+4y)a+(-x+y)b+(x+4y)c`
As the vectors a,b,c are non-coplanar, we can equate their coefficients.
`implies 0=2x+4y`
`implies-3=-x+y`
`implies-2=x+4y`
`x=2,y=-1` is unique solutionn for the above system of equations.
`impliesPQ=2PR-PS`
PQ,PR,PS are coplanar because PQ is a linear combination of PR and PS
`implies`The points P,Q,R,S are also coplanar.
trick for the vectors PQ,PR and PS to be coplanar, we
must have `|(0,-3,-2),(2,-1,1),(4,1,4)|=0` which is true
`therefore`The PQ,PR,PS are coplanar.
Hence, the points P,Q,R,S are also coplanar.
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