Is the vector sum of the unit vectors i ^ and j ^ a unit vector? If no, can you multiply this sum by a scalar number to get a unit vector?

No, the vector sum of the unit vectors i ^ and j ^ is not a unit vector, the vector is written as i ^ + j ^ because the magnitude of the resultant of sum of unit vectors i ^ and j ^ is not one. The magnitude of the resultant sum vector is R = 1 2 + 1 2 = 2 . Yes, we multiply the resultant vector by scalar number 1 2 we can get a unit vector as the magnitude of the resultant will become one.

Let A → = 3 i ^ + 4 j ^ . Write four vectors B → , such that A → ≠ B → but A = B .

Given, A → = 3 i ^ + 4 j ^ A vector B → such that A → ≠ B → means not the same vector as A → , but A = B means the magnitude of A is equal to that of B . There can be a lot of such vectors, by shuffling 3 and 4 with vector i ^ , j ^ and k ^ with changing signs. Some of them are 3 i ^ - 4 j ^ , - 3 i ^ + 4 j ^ , 3 i ^ + 4 k ^ , 3 i ^ - 4 k ^ and many more.

Can you have, A → × B → = A → · B → with A ≠ 0 and B ≠ 0 ? What if one of the two vectors is zero.

No, we cannot have A → × B → = A → · B → with A ≠ 0 and B ≠ 0 . For two quantities to be equal, they must have the same dimensions first. But, in the given question, the left-hand side of the given equation gives a vector quantity as a result, whereas the right-hand side gives a scalar quantity as a result, both can never be equal. The same will go for the condition when one of the vectors is zero.

If A → × B → = 0 , can you say that, ( a ) A → = B → b A → ≠ B → ?

If A → × B → = 0 , then both the vectors are either parallel or antiparallel or one of them is zero or both of them are zero. Anti-parallel means angle between them is 180 ° and for parallel the angle is 0 ° . A → × B → = A B sinθ = 0 ⇒ θ = 180 ° , 0 ° Both the conditions can be satisfied, a A → = B → , the two vectors are equal in magnitude and parallel to each other, since they are parallel they'll satisfy A → × B → = 0 . b A → ≠ B → , if the two vectors are unequal in magnitude and parallel or anti-parallel to each other, they'll satisfy case b as well as A → × B → = 0 .

Let, A → = 5 i ^ - 4 j ^ and B → = - 7 . 5 i ^ + 6 j ^ . Do we have B → = k A → ? Can we say B → A → = k ?

Given, B → = k A → . . . 1 , this is valid like relation F → = m a → and m = F → a → . So, from equation 1 ⇒ - 7 . 5 i ^ + 6 j ^ = k 5 i ^ - 4 j ^ ⇒ k = - 7 . 5 i ^ + 6 j ^ 5 i ^ - 4 j ^ ⇒ k = ± 7 . 5 2 + 6 2 5 2 + 4 2 k = ± 1 . 5 here k = - 1 . 5 satisfies the equation 1 We can't say B → A → = k , because there is no such thing as the division of vectors. But we can say B → A → = k , this is valid.