JEE Main Previous Year Questions (2025): Vectors | Physics for JEE Main & Advanced PDF Download

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JEE Main Previous Year Questions 
(2025): Vectors 
Q1: Two particles are located at equal distance from origin. The position vectors of 
those are represented by ?? ?
= ?? ??ˆ + ?? ?? ?? ˆ + ?? ?? ˆ
 and ?? ?
= ?? ??ˆ - ?? ?? ˆ + ?? ?? ?? ˆ
, respectively. If 
both the vectors are at right angle to each other, the value of ?? -?? is ____ . 
Answer: 3 
Explanation: 
We are given two conditions: 
The particles are at the same distance from the origin. This means that the magnitudes of their 
position vectors are equal. 
The vectors are perpendicular (at right angles) to each other. This means their dot product is 
zero. 
Let's work through these step-by-step. 
? The vectors are given by: 
?? ?
 = 2??ˆ + 3?? ??ˆ + 2?? ˆ
,
?? ??
 = 2??ˆ - 2??ˆ + 4?? ?? ˆ
.
 
? Since the vectors are perpendicular, their dot product must be zero: 
?? ?
· ?? ??
= (2)(2) + (3?? )(-2) + (2)(4?? ) = 4 - 6?? + 8?? = 0. 
This can be rearranged to: 
8?? = 6?? - 4 ? ?? =
3?? - 2
4
  (Equation 1).  
? Since the distances from the origin are equal, the magnitudes of the vectors must be 
equal. Compute the squares of the magnitudes: 
For ?? ?
: |?? ?
|
2
= 2
2
+ (3?? )
2
+ 2
2
= 4 + 9?? 2
+ 4 = 8 + 9?? 2
. 
For ?? ??
: |?? ??
|
2
= 2
2
+ (-2)
2
+ (4?? )
2
= 4 + 4 + 16?? 2
= 8 + 16?? 2
. 
Setting them equal: 
8 + 9?? 2
= 8 + 16?? 2
 ? 9?? 2
= 16?? 2
. 
This simplifies to: 
?? 2
=
9?? 2
16
 ? ?? = ±
3?? 4
  (Equation 2).  
? Now equate the two expressions for ?? from Equation 1 and Equation 2. 
Page 2


JEE Main Previous Year Questions 
(2025): Vectors 
Q1: Two particles are located at equal distance from origin. The position vectors of 
those are represented by ?? ?
= ?? ??ˆ + ?? ?? ?? ˆ + ?? ?? ˆ
 and ?? ?
= ?? ??ˆ - ?? ?? ˆ + ?? ?? ?? ˆ
, respectively. If 
both the vectors are at right angle to each other, the value of ?? -?? is ____ . 
Answer: 3 
Explanation: 
We are given two conditions: 
The particles are at the same distance from the origin. This means that the magnitudes of their 
position vectors are equal. 
The vectors are perpendicular (at right angles) to each other. This means their dot product is 
zero. 
Let's work through these step-by-step. 
? The vectors are given by: 
?? ?
 = 2??ˆ + 3?? ??ˆ + 2?? ˆ
,
?? ??
 = 2??ˆ - 2??ˆ + 4?? ?? ˆ
.
 
? Since the vectors are perpendicular, their dot product must be zero: 
?? ?
· ?? ??
= (2)(2) + (3?? )(-2) + (2)(4?? ) = 4 - 6?? + 8?? = 0. 
This can be rearranged to: 
8?? = 6?? - 4 ? ?? =
3?? - 2
4
  (Equation 1).  
? Since the distances from the origin are equal, the magnitudes of the vectors must be 
equal. Compute the squares of the magnitudes: 
For ?? ?
: |?? ?
|
2
= 2
2
+ (3?? )
2
+ 2
2
= 4 + 9?? 2
+ 4 = 8 + 9?? 2
. 
For ?? ??
: |?? ??
|
2
= 2
2
+ (-2)
2
+ (4?? )
2
= 4 + 4 + 16?? 2
= 8 + 16?? 2
. 
Setting them equal: 
8 + 9?? 2
= 8 + 16?? 2
 ? 9?? 2
= 16?? 2
. 
This simplifies to: 
?? 2
=
9?? 2
16
 ? ?? = ±
3?? 4
  (Equation 2).  
? Now equate the two expressions for ?? from Equation 1 and Equation 2. 
Case 1: Assume ?? =
3?? 4
. 
Then, 
3?? - 2
4
=
3?? 4
 ? 3?? - 2 = 3?? ? -2 = 0,  
which is a contradiction. 
Case 2: Assume ?? = -
3?? 4
. 
Then, 
3?? - 2
4
= -
3?? 4
 ? 3?? - 2 = -3?? .  
Simplify by adding 3?? to both sides: 
6?? - 2 = 0 ? 6?? = 2 ? ?? =
1
3
. 
? Since we found ?? =
1
3
, its reciprocal is: 
?? -1
= 3. 
Thus, the value of ?? -1
 is 3 .