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Question

A unit vector coplanar with ^i+^j+2^k and ^i+2^j+^k and perpendicular to ^i+^j+^k is_________

A
12ˆj+12ˆk

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B
12ˆj12ˆk

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C
12ˆj+12ˆk

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D
None of these
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Solution

The correct option is A
12ˆj+12ˆk


Let the unit vector be xˆi+yˆj+zˆk

then x2+y2+z2=1 (1)

since this vector is coplaner with ˆi+ˆj+2ˆk and ˆi+2ˆj+ˆk

therefore

∣ ∣xyz112121∣ ∣=0

x(14)(12)+z(21)=0

3x+y+z=0

3xyz=0(2)

also this unit vector is perpendicular to ˆi+ˆj+ˆk

so, x+y+z=0(3)

Adding 2 and 3 we get

4x=0

x=0

from 3

x+y+z=0

y+z=0

y=z

putting value of x and y in (1)

x2+y2+z2=1

0+z2+z2=1

2z2=1

z2=12

z=±12

if z=12y=12

and if z=12y=12

thus the required unit vector is

12ˆj+12ˆk or 12ˆj+12ˆk

Therefore the required vector is 12ˆj+12ˆk


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