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 .

E M B I B E

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 2 - Physics and Mathematics>QUESTIONS FOR SHORT ANSWER>Q 5

EASY

JEE Main

IMPORTANT

Earn 100

Can we have physical quantities having magnitude and direction which are not vectors?

Important Questions on Physics and Mathematics

EASY

JEE Main

IMPORTANT

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 2 - Physics and Mathematics>QUESTIONS FOR SHORT ANSWER>Q 6

Which of the following two statements is more appropriate?
$(a)$ Two forces are added using the triangle rule because the force is a vector quantity.

$(b)$ Force is a vector quantity because two forces are added using triangle rule.

EASY

JEE Main

IMPORTANT

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 2 - Physics and Mathematics>QUESTIONS FOR SHORT ANSWER>Q 7

Can you add two vectors representing physical quantities having different dimensions? Can you multiply two vectors representing physical quantities having different dimensions?

EASY

JEE Main

IMPORTANT

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 2 - Physics and Mathematics>QUESTIONS FOR SHORT ANSWER>Q 8

Can a vector have zero components along a line and still have non-zero magnitude?

EASY

JEE Main

IMPORTANT

CONCEPTS OF PHYSICS [VOLUME 1]>Chapter 2 - Physics and Mathematics>QUESTIONS FOR SHORT ANSWER>Q 9

Let