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Question

Which of the following is a unit vector in the direction of ^i+^j+^k.

A
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B
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C
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D
None of these
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Solution

The correct option is C
Magnitude of (^i+^j+^k)=12+12+12=3
Now from scalar multiplication of vectors, we know that if we multiply a vector with a positive scalar, its direction remains unchanged, but its magnitude changes. So if I multiply ^i+^j+^k by 13, what will be its magnitude? Let’s check the magnitude of ^i3+^j3+^k3=(13)2+(13)2+(13)2=1
Now this has magnitude 1 with the direction same as that of ^i+^j+^k
Why the directions of ^i3+^j3+^k3 and ^i+^j+^k are same? Because they are just scalar multiples of each other. And scalar multiplication does not change the direction of a vector but only changes its magnitude.
So ^i3+^j3+^k3 is a unit vector in the direction of ^i+^j+^k.
I have done nothing, just divided the vector by its magnitude to make its magnitude = 1, keeping the direction same as that of original vector.

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