Test: Introductory Vectors - JEE MCQ

We have : 
A= i-3j+2k and B= 3i +6j-7k
So here we want A+B+C= j. -(1) ( j is the unit vector along y axis) 

So A+B = 4i + 3j -5k. - (2)

So by subst.( 2) in( 1) :

4i + 3j -5k. + C= j 
C= j - ( 4i + 3j -5k.)
= j-4i -3j +5k 
= -4i -2j + 5k 

Hence, -4i -2j + 5k is the vector which must be added so that the resultant is a unit vector along y direction.

tanθ= perpendicular/base
tanθ=2/3   as it is with respect to y-axis

Length in XY plane = 

Resolving a Vector into Arbitrary Vectors

To understand why option C is correct, let's first discuss what it means to resolve a vector.

Resolving a vector means breaking it down into its components or constituents. This process allows us to represent a vector in terms of other vectors that have specific properties or characteristics.

In this case, we are considering an arbitrary vector, which means it can be in any direction. According to option C, it is possible to resolve this vector into arbitrary vectors, which means we can represent it in terms of other vectors without any specific constraints.

Now, let's delve into the details of why option C is correct:

1. Arbitrary Vectors

Arbitrary vectors are vectors that do not have any specific properties or characteristics. They can have any direction, magnitude, or orientation. When resolving an arbitrary vector, we can represent it in terms of other vectors without any constraints or limitations. This means we can choose any vectors as long as their resultant is equivalent to the original vector.

2. Original Vector as Resultant

When we resolve an arbitrary vector into arbitrary vectors, we want the sum or resultant of these vectors to be equal to the original vector. In other words, by adding all the arbitrary vectors together, we should be able to recreate the original vector.

This is possible because vectors follow the principles of vector addition. Regardless of the direction or magnitude of the arbitrary vectors, their sum can always be adjusted to match the original vector.

Therefore, option C is correct as it states that an arbitrary vector can be resolved into arbitrary vectors that have the original vector as their resultant. This means we can represent the original vector using any combination of vectors, as long as their sum equals the original vector.

It is important to note that option D, which states that it is not possible to resolve a vector, is incorrect. Vectors can always be resolved into their components or constituents, allowing us to analyze and manipulate them more easily.

  As   therefore velocity of the particle is perpendicular to the position vector.

Magnitude of unit vector = 1     

 By solving we get

⇒ cosθ=0
∴ θ=90º