`hat(i)` and `hat(j)` are unit vectors along x-and y-axis respectively. What is the magnitude and the direction of the vectors `hat(i)+hat(j)` and `hat(i)-hat(j)`? What are the components of a vector `vec(A)=2hat(i)+3hat(j)` along the direction `hat(i)+hat(j)`and `hat(i)-hat(j)`?

Explanation

(i) `hat(i)+hat(j)= sqrt((1)^(2)+(1)^(2) + 2 xx 1 xx1 xx cos90^(@)` = `sqrt(2)`= 1.414units
The vector `|hat(i)-hat(j)|= sqrt((1)^(2)+(2)^(2)-2xx1xx1xxcos90^(@)`= `sqrt(2)=1.414units`
`tantheta =1/1=`, `therefore=45^(@)`
So the vector `hat(i)+hat(j)` makes an angle of `45^(@)` with x-axis.
(ii) `|hat(i)-hat(j)|=sqrt((1)^(2)+(2)^(2)-2xx1xx1xxcos90^(@))`
=`sqrt(2)= 1.414units`
The vector `hat(i)-hat(j)` makes an angle of `-45^(@)` with x-axis.
iii) Let us now determine the component of `vecA=2hat(i)+3hat(j)` in the direction of `hat(i)+hat(j)`.
Let `vecB=hat(i)+hat(j)`
`vecA.vecB=ABcostheta=(Acostheta)B`
So the component of `vecA` in the direction of `vecB`= `(vecA.vecB)/(B)`
`=((2hat(i)+3hat(j)).(hat(i)+hat(j)))/(sqrt(1)^(2)+(1)^(2))` = `(2hat(i).hat(i)+2hat(i).hat(j)+3hat(j).hat(j))/(sqrt(2)=5/sqrt(2)units`
iv) Component of `vec(A)` in the direction of `hat(i)-hat(j)` = `((2hat(i)+3hat(j).(hat(i)-hat(j)`