`hat(i)` and `hat(j)` are unit vectors along x-and y-axes 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

a.`|hat(i)+hat(j)|=sqrt(1^(2)+1^(2)+2xx1xx1xxcos90^(@))`
`=sqrt(2)=1.414` units
` tan theta=1/1=1`
`:.theta =45^(@)`
So, the vector `hat(i)+hat(j)` makes an angle of `45^(@)` with x-axis.
b. `|hat(i)-hat(j)|=|hat(i)+(-hat(j))|`
`=sqrt(1^(2)+(-1)^(2)+2(1)(-1)cos 90^(@))=sqrt2`
`tantheta=1/-1=-1 :.theta=-45^(@)`
The vector `vec(A)=2hat(i)+3hat(j)` makes an angle of `-45^(@)` with x=axis.
Let us know determine the component of `vec(A)=2hat(i)+3hat(j)` in the direction of `hat(i)+hat(j).`
Then, `vec(B)=hat(i)+hat(j)`
`vec(A).vec(B)=AB cos theta=(A cos theta)B`
So component of `vec(A)` in the direction of `vec(B)`
`=(vec(A)*vec(B))/B=((2hat(i)+3hat(j)).(hat(i)+hat(j)))/sqrt(1^(2)+1^(2))=(2.1+3.1)/sqrt(2)=5/sqrt(2)` units
Component of `vec(A)` in the direction of `hat(i)-hat(j)`
`((2hat(i)+3hat(j)).(hat(i)-hat(j)))/sqrt(2)=-1/sqrt(2)` units.
Here,PM is the component of `vec(A)` in the direction of `vec(B)`. So, dot product of two vectors `vec(A)` and `vec(B)` may be defined as the product of the magnitude of `vec(B)` and the component of `vec(A)`in the direction `vec(B)`.