`hat(i)` and `hat(j)` are unit vectors along x-and y-axis respectively. What is the magnitude and direction of the vectors `hat(i) + hat(j)` and `hat(i) - hat(j)` ? What are the components of a vector `A = 2hat(i) + 3hat(j)` along the directions of `hat(i) + 2hat(j)` and `hat(i) - hat(j)` ? [You may use graphical method]

Explanation

(a) Magnitude of `(hat(i) + hat(j)) = |hat(i) + hat(j)| = sqrt((1)^(2) + (1)^(2)) = sqrt(2)`
Let the vector `(hat(i) + hat(j))` (make an angle `theta` with the direction of `hat(i)`, then `cos theta = ((hat(i) + hat(j)).hat(i))/(|hat(i) + hat(j)||hat(i)|)`
`= (1)/((sqrt(2))(1))`
`cos theta = (1)/(sqrt(2)), therefore theta = 45^(@) [(therefore cos theta = vec(A).vec(B))/(AB)]`
Magnitude of `(hat(i) - hat(j)) |hat(i) - hat(j)|`
`= sqrt((1)^(2) + (-1)^(2)) = sqrt(2)`
Similarly, If `theta` is the angle which the vector `(hat(i) - hat(j))` makes with the direction of `hat(i)` then
`cos theta = ((hat(i) - hat(j)).hat(i))/(|hat(i)-hat(j)||hat(i)|) = (1)/(sqrt(2)) = cos 45^(@)` or `theta = 45^(@)`
Here `theta = -45^(@)` with `hat(i)`
(b) Here, `vec(B) = 2hat(i) + 3hat(j)`
To find the component vectors of `vec(A)` along the vectors `(hat(i) + hat(j))` we first find the unit vector along the vector `(hat(i) + hat(j))`. Let a be the unit vector along the direction of vector `(hat(i) + vec(j))`.
Then `hat(a) = (hat(i) + hat(j))/(|hat(i) + hat(j)|) = (hat(i) + hat(j))/(sqrt(1^(2)+1^(2))) = (hat(i) + hat(j))/(sqrt(2))`
`therefore` Component of `vec(B)` along `(hat(i) + hat(j)) = (vec(B).hat(a))`
`hat(a) = (5)/(sqrt(2))((hat(i) + hat(j))/(sqrt(2)))`
`= (5)/(2)(hat(i) + hat(j))`
Let `hat(b)` be the unit vector along the direction of `(hat(i) - hat(j))`
Then, `hat(b) = ((hat(i) - hat(j)))/(|hat(i)-hat(j)|) = (hat(i) - hat(j))/(sqrt(1)^(2) + (-1)^(2)) = 1//sqrt(2)(hat(i) - hat(j))`
Similarly component of `vec(B)` along `(hat(i) - hat(j))` will be `= (vec(B).vec(b))hat(b)`
`= [(2hat(i) + 3hat(j)).(1)/(sqrt(2))(hat(i) - hat(j))](1)/(sqrt(2))(hat(i) - hat(j)) = (1)/(sqrt(2)) (2-3)(1)/(sqrt(2))(hat(i)-hat(j))`
`= -(1)/(2)(hat(i) - hat(j))`