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प्रश्न
`hati "and" hatj` are unit vectors along x- and y-axis respectively. What is the magnitude and direction of the vectors `hati+hatj` and `hati-hatj` ? What are the components of a vector `A = 2hati + 3hatj` along the directions of `hati + hatj` and `hati - hatj` ? [You may use graphical method]
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उत्तर
`hati+hatj = sqrt((1)^2 + (1)^2+2xx1xx1xxcos 90^@) = sqrt2` = 1.414 unit
`tan theta = 1/1 = 1 :. theta = 45^@`
So the vector `hati+hatj` makes an angle `45^@` with x-axis
`|hati-hatj| = sqrt((1)^2 +(2)^2 - 2 xx 1xx1xxcos 90^@`
`= sqrt2` = 1.414 units
The vector `hati -hatj` makess an angle `-45^@` with x-axis
Let us now determined the component of `vecA = 2hati+3hatj` in the direction of `hati+hatj`
Let `vecB = hati+hatj`
`vecA.vecB = AB cos theta = (Acostheta)B`
So the component of `vecA` in the direction of `vecB` = `(vecA.vecB)/B`
`=((2hati+3hatj).(hati+hatj))/sqrt((1)^2+(1)^2) = (2hati.hati+2hati.hatj+3hatj.hati+3hatj.hatj)/sqrt2 = 5/sqrt2 units`
Component of `vecA` in the drection of `hati-hatj = ((2hati+3hatj).(hati-hatj))/sqrt2 = -1/sqrt2` units
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