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The vector ` vec a` has the components `2p` and 1 w.r.t. a rectangular Cartesian system. This system is rotated through a certain angel about the origin in the counterclockwise sense. If, with respect to a new system, ` vec a` has components `(p+1)a n d1` , then `p` is equal to a. `-4` b. `-1//3` c. `1` d. `2`

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The correct Answer is:
2
We have `a=2p hati+hatj`
Let b be the vector obtained from a by rotating the axes. Then, the components of b are p+1 and 1. therefore,
`b=(p+1)hatalpha+hatbeta`
where `hatalpha and hatbeta` are unit vectors along the new axes.
but `|b|=|a|`
`implies 4p^(2)+1=(p+1)^(2)+1`
`implies 3p^(2)-2p-1=0impliesp=1,-(1)/(3)`
`implies p_(1)=1 and p_(2)=-(1)/(3)`
`therefore 3|p_(1)+p_(2)|=3|1-(1)/(3)|=2`.
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