Question

Give two vectors `vec(A)==3hat(i)+4hat(j)` and `vec(B)=hat(i)+hat(j).theta` is the angle between `vec(A)` and `vec(B)`. Which of the following statements is/are correct?
A. `|vec(A)|cos theta((hat(i)+hat(j))/(sqrt(2)))` is the component of `vec(A)` along `vec(B)`.
B. `|vec(A)|sin theta((hat(i)-hat(j))/sqrt(2))` is the component of `vec(A)` perpendicular to `vec(B)`.
C. `|vec(A)|cos theta((hat(i)-hat(j))/(sqrt(2)))` is the component of `vec(A)` along `vec(B)`.
D. `|vec(A)|sin theta((hat(i)+hat(j))/(sqrt(2)))` is the component of `vec(A)` perpendicular to `vec(B)`.

Answer 1

Correct Answer - A::B
Component of `vec(A)` along `vec(B)` is `|vec(A)|cos theta hat(B)` for `theta` being the angle between the vectors.
Also `vec(B)=(hat(i)+hat(j))/sqrt(2)`.So choice (a) is correct.
The vector `(hat(i)-hat(j))` is perpendicular to the vector `(hat(i)+hat(j))`
So the other resolved component is `|vec(A)| sin theta((hat(i)-hat(j))/sqrt(2))`