Question

`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]

Answer 1

`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