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Give two vectors vec(A)==3hat(i)+4hat(j)...

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)`.

Text Solution

Verified by Experts

The correct Answer is:
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))`
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