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Let a=3hat(i) +2hat(j) +2hat(k) " and "...

Let `a=3hat(i) +2hat(j) +2hat(k) " and " b=hat(i) +2hat(j) -2hat(k)` be two vectors. If a vectors perpendicular to both the vectors a+b and a-b has the megnitude 12 ,then one such vectors is

A

`4(2hat(i)+2hat(j)+hat(k))`

B

`4(2hat(i)-2hat(j)-hat(k))`

C

`4(2hat(i)+2hat(j)-hat(k))`

D

`4(-2hat(i)-2hat(j)+hat(k))`

Text Solution

Verified by Experts

The correct Answer is:
B
Given vectors are
` a= 3hat(i) + 2hat(j) + 2hat(k) " and " b= hat(j) + 2hat(j) - 2hat(k)`
Now vectors a+b `=4hat(j) " and " a-b =2hat(i) +4hat(k)`
`:. `A vector which is perpendicular to both the vectors a+b and a-b is
` (a+b) xx (a-b) = |{:(hat(i) ,,hat(j) ,,hat(k) ),(4,,4,,0),(2,,0,,4):}|`
`=hat(i) (16) -hat(j) (16) +hat(k)(-8) =8(2hat(i) -2hat(j) -hat(k))`
Then the required vector along `(a+b) xx (a-b)` having magnitude 12 is
`+- 12 xx (8(2hat(i) -2hat(j) -hat(k)))/(8xx sqrt(4+4+1)) = +- 4(2hat(i) -2hat(j) - hat(k))`
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