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Page 1 JEE Mains Previous Year Questions (2021-2024): Vector Algebra 2024 Q1: Let ??? = -?? ??ˆ + ??ˆ - ?? ?? ˆ ,?? = ??ˆ + ?? ??ˆ - ?? ?? ˆ and ?? = ( ( ??? × ?? )× ??ˆ)× ??ˆ)× ??ˆ. Then ??? · ( -??ˆ + ??ˆ + ?? ˆ ) is equal to : (A) -12 (B) -10 (C) -13 (D) -15 [JEE Main 2024 (Online) 1st February Morning Shift] Ans: (a) ?? = -5 · ??ˆ + ??ˆ - 3?? ˆ ,?? ? = ??ˆ + 2??ˆ - 4?? ˆ ?? = ( ( ( ?? × ?? ? )× ??ˆ)× ??ˆ)× ??ˆ = ( ( ( ?? · ??ˆ) ?? ? - ( ?? ? · ??ˆ) ?? )× ??ˆ)× ??ˆ = ( ( -5?? ? - ?? )× ??ˆ)× ??ˆ = ( ( -11??ˆ + 23?? ˆ )× ??ˆ)× ??ˆ = ( 11?? ˆ + 23??ˆ)× ??ˆ = ( 11??ˆ - 23?? ˆ ) ?? · ( -??ˆ + ??ˆ + ?? ˆ )= 0 + 11- 23 = -12 Q2: Let ??? ? = ?? ??ˆ + ??ˆ - ?? ?? ˆ ,?? ? ? = ?? ??ˆ + ??ˆ + ?? ?? ˆ and ??? = ??ˆ - ?? ??ˆ + ?? ?? ˆ be three vectors. If a vectors ??? ? satisfies ??? ? × ?? ? ? = ??? × ?? ? ? and ??? ? · ??? ? = ?? , then ??? ? · ( ??ˆ - ??ˆ - ?? ˆ ) is equal to (A)24 (B) 32 (C) 36 (D )28 [JEE Main 2024 (Online) 31st January Morning Shift] Ans: (b) ?? × ?? ? - ?? × ?? ? = 0 ? ( ?? - ?? )× ?? ? = 0 ? ?? - ?? = ?? ?? ? ? ?? = ?? + ?? ?? ? Now, p ? · a ? = 0 (given) Page 2 JEE Mains Previous Year Questions (2021-2024): Vector Algebra 2024 Q1: Let ??? = -?? ??ˆ + ??ˆ - ?? ?? ˆ ,?? = ??ˆ + ?? ??ˆ - ?? ?? ˆ and ?? = ( ( ??? × ?? )× ??ˆ)× ??ˆ)× ??ˆ. Then ??? · ( -??ˆ + ??ˆ + ?? ˆ ) is equal to : (A) -12 (B) -10 (C) -13 (D) -15 [JEE Main 2024 (Online) 1st February Morning Shift] Ans: (a) ?? = -5 · ??ˆ + ??ˆ - 3?? ˆ ,?? ? = ??ˆ + 2??ˆ - 4?? ˆ ?? = ( ( ( ?? × ?? ? )× ??ˆ)× ??ˆ)× ??ˆ = ( ( ( ?? · ??ˆ) ?? ? - ( ?? ? · ??ˆ) ?? )× ??ˆ)× ??ˆ = ( ( -5?? ? - ?? )× ??ˆ)× ??ˆ = ( ( -11??ˆ + 23?? ˆ )× ??ˆ)× ??ˆ = ( 11?? ˆ + 23??ˆ)× ??ˆ = ( 11??ˆ - 23?? ˆ ) ?? · ( -??ˆ + ??ˆ + ?? ˆ )= 0 + 11- 23 = -12 Q2: Let ??? ? = ?? ??ˆ + ??ˆ - ?? ?? ˆ ,?? ? ? = ?? ??ˆ + ??ˆ + ?? ?? ˆ and ??? = ??ˆ - ?? ??ˆ + ?? ?? ˆ be three vectors. If a vectors ??? ? satisfies ??? ? × ?? ? ? = ??? × ?? ? ? and ??? ? · ??? ? = ?? , then ??? ? · ( ??ˆ - ??ˆ - ?? ˆ ) is equal to (A)24 (B) 32 (C) 36 (D )28 [JEE Main 2024 (Online) 31st January Morning Shift] Ans: (b) ?? × ?? ? - ?? × ?? ? = 0 ? ( ?? - ?? )× ?? ? = 0 ? ?? - ?? = ?? ?? ? ? ?? = ?? + ?? ?? ? Now, p ? · a ? = 0 (given) So, c · a ? + ?? a ? · b ? = 0 ( 3 - 3 - 8)+ ?? ( 12+ 1 - 14)= 0 ?? = -8 p ? = c - 8b ? p ? = -31i ˆ - 11j ˆ - 52k ˆ So, p ? · ( i ˆ - j ˆ - k ˆ ) = -31+ 11+ 52 = 32 Q3: The distance of the point ?? ( ?? ,?? ,-?? ) form the line passing through the point ?? ( ?? ,-?? ,?? ) and perpendicular to the lines ??? = ( -?? ??ˆ + ?? ?? ˆ )+ ?? ( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) ,?? ? R and ??? = ( ??ˆ - ?? ??ˆ + ?? ˆ )+ ?? ( -??ˆ + ?? ??ˆ + ?? ?? ˆ ) ,?? ? R is : (A) v???? (B) v???? (C) v???? (D) v???? [JEE Main 2024 (Online) 31st January Morning Shift] Ans: (a) A vector in the direction of the required line can be obtained by cross product of | ??ˆ ??ˆ ?? ˆ 2 3 5 -1 3 2 | = -9??ˆ - 9??ˆ + 9?? ˆ Required line r = ( 5i ˆ - 4j ˆ + 3k ˆ )+ ?? ' ( -9i ˆ - 9j ˆ + 9k ˆ ) r = ( 5i ˆ - 4j ˆ + 3k ˆ )+ ?? ( i ˆ + j ˆ - k ˆ ) Now distance of ( 0,2,-2) Page 3 JEE Mains Previous Year Questions (2021-2024): Vector Algebra 2024 Q1: Let ??? = -?? ??ˆ + ??ˆ - ?? ?? ˆ ,?? = ??ˆ + ?? ??ˆ - ?? ?? ˆ and ?? = ( ( ??? × ?? )× ??ˆ)× ??ˆ)× ??ˆ. Then ??? · ( -??ˆ + ??ˆ + ?? ˆ ) is equal to : (A) -12 (B) -10 (C) -13 (D) -15 [JEE Main 2024 (Online) 1st February Morning Shift] Ans: (a) ?? = -5 · ??ˆ + ??ˆ - 3?? ˆ ,?? ? = ??ˆ + 2??ˆ - 4?? ˆ ?? = ( ( ( ?? × ?? ? )× ??ˆ)× ??ˆ)× ??ˆ = ( ( ( ?? · ??ˆ) ?? ? - ( ?? ? · ??ˆ) ?? )× ??ˆ)× ??ˆ = ( ( -5?? ? - ?? )× ??ˆ)× ??ˆ = ( ( -11??ˆ + 23?? ˆ )× ??ˆ)× ??ˆ = ( 11?? ˆ + 23??ˆ)× ??ˆ = ( 11??ˆ - 23?? ˆ ) ?? · ( -??ˆ + ??ˆ + ?? ˆ )= 0 + 11- 23 = -12 Q2: Let ??? ? = ?? ??ˆ + ??ˆ - ?? ?? ˆ ,?? ? ? = ?? ??ˆ + ??ˆ + ?? ?? ˆ and ??? = ??ˆ - ?? ??ˆ + ?? ?? ˆ be three vectors. If a vectors ??? ? satisfies ??? ? × ?? ? ? = ??? × ?? ? ? and ??? ? · ??? ? = ?? , then ??? ? · ( ??ˆ - ??ˆ - ?? ˆ ) is equal to (A)24 (B) 32 (C) 36 (D )28 [JEE Main 2024 (Online) 31st January Morning Shift] Ans: (b) ?? × ?? ? - ?? × ?? ? = 0 ? ( ?? - ?? )× ?? ? = 0 ? ?? - ?? = ?? ?? ? ? ?? = ?? + ?? ?? ? Now, p ? · a ? = 0 (given) So, c · a ? + ?? a ? · b ? = 0 ( 3 - 3 - 8)+ ?? ( 12+ 1 - 14)= 0 ?? = -8 p ? = c - 8b ? p ? = -31i ˆ - 11j ˆ - 52k ˆ So, p ? · ( i ˆ - j ˆ - k ˆ ) = -31+ 11+ 52 = 32 Q3: The distance of the point ?? ( ?? ,?? ,-?? ) form the line passing through the point ?? ( ?? ,-?? ,?? ) and perpendicular to the lines ??? = ( -?? ??ˆ + ?? ?? ˆ )+ ?? ( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) ,?? ? R and ??? = ( ??ˆ - ?? ??ˆ + ?? ˆ )+ ?? ( -??ˆ + ?? ??ˆ + ?? ?? ˆ ) ,?? ? R is : (A) v???? (B) v???? (C) v???? (D) v???? [JEE Main 2024 (Online) 31st January Morning Shift] Ans: (a) A vector in the direction of the required line can be obtained by cross product of | ??ˆ ??ˆ ?? ˆ 2 3 5 -1 3 2 | = -9??ˆ - 9??ˆ + 9?? ˆ Required line r = ( 5i ˆ - 4j ˆ + 3k ˆ )+ ?? ' ( -9i ˆ - 9j ˆ + 9k ˆ ) r = ( 5i ˆ - 4j ˆ + 3k ˆ )+ ?? ( i ˆ + j ˆ - k ˆ ) Now distance of ( 0,2,-2) P.V. of P = ( 5 + ?? ) ??ˆ + ( ?? - 4) ??ˆ + ( 3 - ?? ) ?? ˆ AP ????? = ( 5 + ?? ) ??ˆ + ( ?? - 6) ??ˆ + ( 5 - ?? ) ?? ˆ AP ????? · ( i ˆ + j ˆ - k ˆ )= 0 5 + ?? + ?? - 6 - 5 + ?? = 0 ?? = 2 |AP ????? | = v49 + 16 + 9 |AP ????? | = v74 Q4: Let ?? ?? :??? = ( ??ˆ - ??ˆ + ?? ?? ˆ )+ ?? ( ??ˆ - ??ˆ + ?? ?? ˆ ) ,?? ? R, ?? ?? :??? = ( ??ˆ - ?? ˆ )+ ?? ( ?? ??ˆ + ??ˆ + ?? ?? ˆ ) ,?? ? R, and ?? ?? :??? = ?? ( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) ,?? ? R be three lines such that ?? ?? is perpendicular to ?? ?? and ?? ?? is perpendicular to both ?? ?? and ?? ?? . Then, the point which lies on ?? ?? is (A) ( ?? ,?? ,-?? ) (B) ( ?? ,-?? ,?? ) (C) ( -?? ,?? ,?? ) (D) ( -,?? - ?? ,?? ) [JEE Main 2024 (Online) 30th January Evening Shift] Ans: (c) L 1 ? L 2 L 3 ? L 1 , L 2 3 - 1 + 2P = 0 P = -1 | i ˆ j ˆ k ˆ 1 -1 2 3 1 -1 | = -i ˆ + 7j ˆ + 4k ˆ ? ( -?? ,7?? ,4?? ) will lie on L 3 For ?? = 1 the point will be ( -1,7,4) Q5: Let ??? ? = ??ˆ + ?? ??ˆ + ?? ?? ˆ ,?? ,?? ? R. Let a vector ?? ? ? be such that the angle between ??? ? and ?? ? ? is ?? ?? and |?? ? ? | ?? = ?? . If ??? ? · ?? ? ? = ?? v?? , then the value of ( ?? ?? + ?? ?? ) |??? ? × ?? ? ? | ?? is equal to A. 85 B. 90 C. 75 D. 95 [JEE Main 2024 (Online) 30th January Evening Shift] Ans: (b) |b ? | 2 = 6;|a ? ||b ? |cos ?? = 3v2 |a ? | 2 |b ? | 2 cos 2 ?? = 18 |a ? | 2 = 6 Also 1 + ?? 2 + ?? 2 = 6 ?? 2 + ?? 2 = 5 to find Page 4 JEE Mains Previous Year Questions (2021-2024): Vector Algebra 2024 Q1: Let ??? = -?? ??ˆ + ??ˆ - ?? ?? ˆ ,?? = ??ˆ + ?? ??ˆ - ?? ?? ˆ and ?? = ( ( ??? × ?? )× ??ˆ)× ??ˆ)× ??ˆ. Then ??? · ( -??ˆ + ??ˆ + ?? ˆ ) is equal to : (A) -12 (B) -10 (C) -13 (D) -15 [JEE Main 2024 (Online) 1st February Morning Shift] Ans: (a) ?? = -5 · ??ˆ + ??ˆ - 3?? ˆ ,?? ? = ??ˆ + 2??ˆ - 4?? ˆ ?? = ( ( ( ?? × ?? ? )× ??ˆ)× ??ˆ)× ??ˆ = ( ( ( ?? · ??ˆ) ?? ? - ( ?? ? · ??ˆ) ?? )× ??ˆ)× ??ˆ = ( ( -5?? ? - ?? )× ??ˆ)× ??ˆ = ( ( -11??ˆ + 23?? ˆ )× ??ˆ)× ??ˆ = ( 11?? ˆ + 23??ˆ)× ??ˆ = ( 11??ˆ - 23?? ˆ ) ?? · ( -??ˆ + ??ˆ + ?? ˆ )= 0 + 11- 23 = -12 Q2: Let ??? ? = ?? ??ˆ + ??ˆ - ?? ?? ˆ ,?? ? ? = ?? ??ˆ + ??ˆ + ?? ?? ˆ and ??? = ??ˆ - ?? ??ˆ + ?? ?? ˆ be three vectors. If a vectors ??? ? satisfies ??? ? × ?? ? ? = ??? × ?? ? ? and ??? ? · ??? ? = ?? , then ??? ? · ( ??ˆ - ??ˆ - ?? ˆ ) is equal to (A)24 (B) 32 (C) 36 (D )28 [JEE Main 2024 (Online) 31st January Morning Shift] Ans: (b) ?? × ?? ? - ?? × ?? ? = 0 ? ( ?? - ?? )× ?? ? = 0 ? ?? - ?? = ?? ?? ? ? ?? = ?? + ?? ?? ? Now, p ? · a ? = 0 (given) So, c · a ? + ?? a ? · b ? = 0 ( 3 - 3 - 8)+ ?? ( 12+ 1 - 14)= 0 ?? = -8 p ? = c - 8b ? p ? = -31i ˆ - 11j ˆ - 52k ˆ So, p ? · ( i ˆ - j ˆ - k ˆ ) = -31+ 11+ 52 = 32 Q3: The distance of the point ?? ( ?? ,?? ,-?? ) form the line passing through the point ?? ( ?? ,-?? ,?? ) and perpendicular to the lines ??? = ( -?? ??ˆ + ?? ?? ˆ )+ ?? ( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) ,?? ? R and ??? = ( ??ˆ - ?? ??ˆ + ?? ˆ )+ ?? ( -??ˆ + ?? ??ˆ + ?? ?? ˆ ) ,?? ? R is : (A) v???? (B) v???? (C) v???? (D) v???? [JEE Main 2024 (Online) 31st January Morning Shift] Ans: (a) A vector in the direction of the required line can be obtained by cross product of | ??ˆ ??ˆ ?? ˆ 2 3 5 -1 3 2 | = -9??ˆ - 9??ˆ + 9?? ˆ Required line r = ( 5i ˆ - 4j ˆ + 3k ˆ )+ ?? ' ( -9i ˆ - 9j ˆ + 9k ˆ ) r = ( 5i ˆ - 4j ˆ + 3k ˆ )+ ?? ( i ˆ + j ˆ - k ˆ ) Now distance of ( 0,2,-2) P.V. of P = ( 5 + ?? ) ??ˆ + ( ?? - 4) ??ˆ + ( 3 - ?? ) ?? ˆ AP ????? = ( 5 + ?? ) ??ˆ + ( ?? - 6) ??ˆ + ( 5 - ?? ) ?? ˆ AP ????? · ( i ˆ + j ˆ - k ˆ )= 0 5 + ?? + ?? - 6 - 5 + ?? = 0 ?? = 2 |AP ????? | = v49 + 16 + 9 |AP ????? | = v74 Q4: Let ?? ?? :??? = ( ??ˆ - ??ˆ + ?? ?? ˆ )+ ?? ( ??ˆ - ??ˆ + ?? ?? ˆ ) ,?? ? R, ?? ?? :??? = ( ??ˆ - ?? ˆ )+ ?? ( ?? ??ˆ + ??ˆ + ?? ?? ˆ ) ,?? ? R, and ?? ?? :??? = ?? ( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) ,?? ? R be three lines such that ?? ?? is perpendicular to ?? ?? and ?? ?? is perpendicular to both ?? ?? and ?? ?? . Then, the point which lies on ?? ?? is (A) ( ?? ,?? ,-?? ) (B) ( ?? ,-?? ,?? ) (C) ( -?? ,?? ,?? ) (D) ( -,?? - ?? ,?? ) [JEE Main 2024 (Online) 30th January Evening Shift] Ans: (c) L 1 ? L 2 L 3 ? L 1 , L 2 3 - 1 + 2P = 0 P = -1 | i ˆ j ˆ k ˆ 1 -1 2 3 1 -1 | = -i ˆ + 7j ˆ + 4k ˆ ? ( -?? ,7?? ,4?? ) will lie on L 3 For ?? = 1 the point will be ( -1,7,4) Q5: Let ??? ? = ??ˆ + ?? ??ˆ + ?? ?? ˆ ,?? ,?? ? R. Let a vector ?? ? ? be such that the angle between ??? ? and ?? ? ? is ?? ?? and |?? ? ? | ?? = ?? . If ??? ? · ?? ? ? = ?? v?? , then the value of ( ?? ?? + ?? ?? ) |??? ? × ?? ? ? | ?? is equal to A. 85 B. 90 C. 75 D. 95 [JEE Main 2024 (Online) 30th January Evening Shift] Ans: (b) |b ? | 2 = 6;|a ? ||b ? |cos ?? = 3v2 |a ? | 2 |b ? | 2 cos 2 ?? = 18 |a ? | 2 = 6 Also 1 + ?? 2 + ?? 2 = 6 ?? 2 + ?? 2 = 5 to find ( ?? 2 + ?? 2 ) |?? | 2 |?? ? | 2 sin 2 ?? = ( 5) ( 6) ( 6)( 1 2 ) = 90 Q6: Let ?? and ?? ? be two vectors such that |?? ? | = 1 and |?? ? × ?? | = 2. Then |( ?? ? × ?? )- ?? ? | 2 is equal to (A) 1 (B) 3 (C) 5 (D) 4 [JEE Main 2024 (Online) 30th January Evening Shift] Ans: (c) |?? ? | = 1&|?? ? × ?? | = 2 ( ?? ? × ?? )· ?? ? = ?? ? · ( ?? ? × ?? )= 0 |( ?? ? × ?? )- ?? ? | 2 = |?? ? × ?? | 2 + |?? ? | 2 = 4 + 1 = 5 Q7: Let ??? = ?? ?? ??ˆ + ?? ?? ??ˆ + ?? ?? ?? ˆ and ?? = ?? ?? ??ˆ ˆ + ?? ?? ??ˆ + ?? ?? ?? ˆ be two vectors such that |??? | = ?? ,??? ? · ?? ? ? = ?? and |?? ? ? | = ?? . If ??? = ?? ( ??? ? × ?? ? ? )- ?? ?? ? ? , then the angle between ?? ? ? and ??? is equal to: (A) ?????? -?? (- ?? v?? ) (B) ?????? -?? ( ?? ?? ) (C) ?????? -?? ( ?? v?? ) (D) ?????? -?? ( - v?? ?? ) [JEE Main 2024 (Online) 30th January Morning Shift] Ans: (d) Given |?? | = 1,|?? ? | = 4,?? · ?? ? = 2 ?? = 2( ?? × ?? ? )- 3?? ? Dot product with a ? on both sides ?? . ?? ?? = -6… Dot product with ?? ? on both sides bc ???? = -48 …( 2) c · c = 4a ? × b ? | 2 + 9|b ? | 2 |c | 2 = 4[|a| 2 | b| 2 - ( a · b ? ) 2 ] + 9|b ? | 2 |c | 2 = 4[( 1) ( 4) 2 - ( 4) ] + 9( 16) |c | 2 = 4[12] + 144 |c | 2 = 48 + 144 |c | 2 = 192 Page 5 JEE Mains Previous Year Questions (2021-2024): Vector Algebra 2024 Q1: Let ??? = -?? ??ˆ + ??ˆ - ?? ?? ˆ ,?? = ??ˆ + ?? ??ˆ - ?? ?? ˆ and ?? = ( ( ??? × ?? )× ??ˆ)× ??ˆ)× ??ˆ. Then ??? · ( -??ˆ + ??ˆ + ?? ˆ ) is equal to : (A) -12 (B) -10 (C) -13 (D) -15 [JEE Main 2024 (Online) 1st February Morning Shift] Ans: (a) ?? = -5 · ??ˆ + ??ˆ - 3?? ˆ ,?? ? = ??ˆ + 2??ˆ - 4?? ˆ ?? = ( ( ( ?? × ?? ? )× ??ˆ)× ??ˆ)× ??ˆ = ( ( ( ?? · ??ˆ) ?? ? - ( ?? ? · ??ˆ) ?? )× ??ˆ)× ??ˆ = ( ( -5?? ? - ?? )× ??ˆ)× ??ˆ = ( ( -11??ˆ + 23?? ˆ )× ??ˆ)× ??ˆ = ( 11?? ˆ + 23??ˆ)× ??ˆ = ( 11??ˆ - 23?? ˆ ) ?? · ( -??ˆ + ??ˆ + ?? ˆ )= 0 + 11- 23 = -12 Q2: Let ??? ? = ?? ??ˆ + ??ˆ - ?? ?? ˆ ,?? ? ? = ?? ??ˆ + ??ˆ + ?? ?? ˆ and ??? = ??ˆ - ?? ??ˆ + ?? ?? ˆ be three vectors. If a vectors ??? ? satisfies ??? ? × ?? ? ? = ??? × ?? ? ? and ??? ? · ??? ? = ?? , then ??? ? · ( ??ˆ - ??ˆ - ?? ˆ ) is equal to (A)24 (B) 32 (C) 36 (D )28 [JEE Main 2024 (Online) 31st January Morning Shift] Ans: (b) ?? × ?? ? - ?? × ?? ? = 0 ? ( ?? - ?? )× ?? ? = 0 ? ?? - ?? = ?? ?? ? ? ?? = ?? + ?? ?? ? Now, p ? · a ? = 0 (given) So, c · a ? + ?? a ? · b ? = 0 ( 3 - 3 - 8)+ ?? ( 12+ 1 - 14)= 0 ?? = -8 p ? = c - 8b ? p ? = -31i ˆ - 11j ˆ - 52k ˆ So, p ? · ( i ˆ - j ˆ - k ˆ ) = -31+ 11+ 52 = 32 Q3: The distance of the point ?? ( ?? ,?? ,-?? ) form the line passing through the point ?? ( ?? ,-?? ,?? ) and perpendicular to the lines ??? = ( -?? ??ˆ + ?? ?? ˆ )+ ?? ( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) ,?? ? R and ??? = ( ??ˆ - ?? ??ˆ + ?? ˆ )+ ?? ( -??ˆ + ?? ??ˆ + ?? ?? ˆ ) ,?? ? R is : (A) v???? (B) v???? (C) v???? (D) v???? [JEE Main 2024 (Online) 31st January Morning Shift] Ans: (a) A vector in the direction of the required line can be obtained by cross product of | ??ˆ ??ˆ ?? ˆ 2 3 5 -1 3 2 | = -9??ˆ - 9??ˆ + 9?? ˆ Required line r = ( 5i ˆ - 4j ˆ + 3k ˆ )+ ?? ' ( -9i ˆ - 9j ˆ + 9k ˆ ) r = ( 5i ˆ - 4j ˆ + 3k ˆ )+ ?? ( i ˆ + j ˆ - k ˆ ) Now distance of ( 0,2,-2) P.V. of P = ( 5 + ?? ) ??ˆ + ( ?? - 4) ??ˆ + ( 3 - ?? ) ?? ˆ AP ????? = ( 5 + ?? ) ??ˆ + ( ?? - 6) ??ˆ + ( 5 - ?? ) ?? ˆ AP ????? · ( i ˆ + j ˆ - k ˆ )= 0 5 + ?? + ?? - 6 - 5 + ?? = 0 ?? = 2 |AP ????? | = v49 + 16 + 9 |AP ????? | = v74 Q4: Let ?? ?? :??? = ( ??ˆ - ??ˆ + ?? ?? ˆ )+ ?? ( ??ˆ - ??ˆ + ?? ?? ˆ ) ,?? ? R, ?? ?? :??? = ( ??ˆ - ?? ˆ )+ ?? ( ?? ??ˆ + ??ˆ + ?? ?? ˆ ) ,?? ? R, and ?? ?? :??? = ?? ( ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ ) ,?? ? R be three lines such that ?? ?? is perpendicular to ?? ?? and ?? ?? is perpendicular to both ?? ?? and ?? ?? . Then, the point which lies on ?? ?? is (A) ( ?? ,?? ,-?? ) (B) ( ?? ,-?? ,?? ) (C) ( -?? ,?? ,?? ) (D) ( -,?? - ?? ,?? ) [JEE Main 2024 (Online) 30th January Evening Shift] Ans: (c) L 1 ? L 2 L 3 ? L 1 , L 2 3 - 1 + 2P = 0 P = -1 | i ˆ j ˆ k ˆ 1 -1 2 3 1 -1 | = -i ˆ + 7j ˆ + 4k ˆ ? ( -?? ,7?? ,4?? ) will lie on L 3 For ?? = 1 the point will be ( -1,7,4) Q5: Let ??? ? = ??ˆ + ?? ??ˆ + ?? ?? ˆ ,?? ,?? ? R. Let a vector ?? ? ? be such that the angle between ??? ? and ?? ? ? is ?? ?? and |?? ? ? | ?? = ?? . If ??? ? · ?? ? ? = ?? v?? , then the value of ( ?? ?? + ?? ?? ) |??? ? × ?? ? ? | ?? is equal to A. 85 B. 90 C. 75 D. 95 [JEE Main 2024 (Online) 30th January Evening Shift] Ans: (b) |b ? | 2 = 6;|a ? ||b ? |cos ?? = 3v2 |a ? | 2 |b ? | 2 cos 2 ?? = 18 |a ? | 2 = 6 Also 1 + ?? 2 + ?? 2 = 6 ?? 2 + ?? 2 = 5 to find ( ?? 2 + ?? 2 ) |?? | 2 |?? ? | 2 sin 2 ?? = ( 5) ( 6) ( 6)( 1 2 ) = 90 Q6: Let ?? and ?? ? be two vectors such that |?? ? | = 1 and |?? ? × ?? | = 2. Then |( ?? ? × ?? )- ?? ? | 2 is equal to (A) 1 (B) 3 (C) 5 (D) 4 [JEE Main 2024 (Online) 30th January Evening Shift] Ans: (c) |?? ? | = 1&|?? ? × ?? | = 2 ( ?? ? × ?? )· ?? ? = ?? ? · ( ?? ? × ?? )= 0 |( ?? ? × ?? )- ?? ? | 2 = |?? ? × ?? | 2 + |?? ? | 2 = 4 + 1 = 5 Q7: Let ??? = ?? ?? ??ˆ + ?? ?? ??ˆ + ?? ?? ?? ˆ and ?? = ?? ?? ??ˆ ˆ + ?? ?? ??ˆ + ?? ?? ?? ˆ be two vectors such that |??? | = ?? ,??? ? · ?? ? ? = ?? and |?? ? ? | = ?? . If ??? = ?? ( ??? ? × ?? ? ? )- ?? ?? ? ? , then the angle between ?? ? ? and ??? is equal to: (A) ?????? -?? (- ?? v?? ) (B) ?????? -?? ( ?? ?? ) (C) ?????? -?? ( ?? v?? ) (D) ?????? -?? ( - v?? ?? ) [JEE Main 2024 (Online) 30th January Morning Shift] Ans: (d) Given |?? | = 1,|?? ? | = 4,?? · ?? ? = 2 ?? = 2( ?? × ?? ? )- 3?? ? Dot product with a ? on both sides ?? . ?? ?? = -6… Dot product with ?? ? on both sides bc ???? = -48 …( 2) c · c = 4a ? × b ? | 2 + 9|b ? | 2 |c | 2 = 4[|a| 2 | b| 2 - ( a · b ? ) 2 ] + 9|b ? | 2 |c | 2 = 4[( 1) ( 4) 2 - ( 4) ] + 9( 16) |c | 2 = 4[12] + 144 |c | 2 = 48 + 144 |c | 2 = 192 ? cos ?? = b ? · c |b ? c | ??????? ? cos ?? = -48 v192· 4 ? cos ?? = -48 8v3 · 4 ? cos ?? = -3 2v3 ? cos ?? = -v3 2 ? ?? = cos -1 ( -v3 2 ) Q8: Let a unit vector ?? ˆ = ?? ??ˆ + ?? ??ˆ + ?? ?? ˆ make angles ?? ?? , ?? ?? and ?? ?? ?? with the vectors ?? v?? ??ˆ + ?? v?? ?? ˆ , ?? v?? ??ˆ + ?? v?? ?? ˆ and ?? v?? ??ˆ + ?? v?? ??ˆ respectively. If ??? ? = ?? v?? ??ˆ + ?? v?? ??ˆ + ?? v?? ?? ˆ then |??ˆ - ??? ? | ?? is equal to A. ???? ?? B. ?? ?? C. 7 D. 9 [JEE Main 2024 (Online) 29th January Evening Shift] Ans: (b) Unit vector u ˆ = xi ˆ + yj ˆ + zk p 1 ???? = 1 v2 i ˆ + 1 v2 k ˆ ,p ? 2 = 1 v2 j ˆ + 1 v2 k ˆ p ? 3 = 1 v2 i ˆ + 1 v2 j ˆ Now angle between u ˆ and p ? 1 = ?? 2 u ˆ · p ? 1 = 0 ? x v2 + z v2 = 0 ? x + z = 0…… (i) Angle between u ˆ and p ? 2 = ?? 3 u ˆ · p 2 ???? = |u ˆ|· |p 2 ???? |cos ?? 3 ? ?? v2 + ?? v2 = 1 2 ? ?? + ?? = 1 v2 Angle between u ˆ and p ? 3 = 2?? 3 u ˆ · p ? 3 = |u ˆ|· |P ? ? 3|cos 2?? 3 ? ?? v2 + 4 v2 = -1 2 ? ?? + ?? = -1 v2 …..Read More
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FAQs on Vector Algebra & Three Dimensional Geometry: JEE Mains Previous Year Questions (2021-2024) - Mathematics (Maths) for JEE Main & Advanced
| 1. How to find the angle between two vectors in three-dimensional space? |  |
Ans. To find the angle between two vectors in three-dimensional space, you can use the formula: cosθ = (a · b) / (|a||b|), where a and b are the two vectors, · denotes the dot product, and |a| and |b| are the magnitudes of the vectors.
| 2. How to determine if two vectors are orthogonal in three-dimensional space? | |
Ans. Two vectors are orthogonal in three-dimensional space if their dot product is equal to zero. In other words, if a · b = 0, where a and b are the vectors, then they are orthogonal.
| 3. How to find the distance between a point and a plane in three-dimensional space? | |
Ans. To find the distance between a point and a plane in three-dimensional space, you can use the formula: d = |ax + by + cz + d| / √(a^2 + b^2 + c^2), where (a, b, c) is the normal vector to the plane, (x, y, z) is the coordinates of the point, and d is the constant term in the plane equation ax + by + cz + d = 0.
| 4. How to determine if three points are collinear in three-dimensional space? | |
Ans. Three points are collinear in three-dimensional space if the vectors formed by connecting them are linearly dependent. This means that the determinant of the matrix formed by the coordinates of the points should be zero.
| 5. How to find the equation of a plane passing through a point and perpendicular to a given vector in three-dimensional space? | |
Ans. To find the equation of a plane passing through a point and perpendicular to a given vector in three-dimensional space, you can use the formula: a(x - x₁) + b(y - y₁) + c(z - z₁) = 0, where (a, b, c) is the given vector, and (x₁, y₁, z₁) is the point the plane passes through.
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