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Let a, b, c be non coplanar unit vectors, making an angle of 60º with each other. If a *b+ b* c =pa qb rc , then the value of 2(p^2 q^2 r^2) is equal to * = here is for cross product?
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?Let a, b, c be non coplanar unit vectors, making an angle of 60º with...
Problem
Let a, b, c be non-coplanar unit vectors, making an angle of 60º with each other. If a * b b * c = pa qb rc, then the value of 2(p^2 q^2 r^2) is equal to what?
Solution
Given:
a, b, c are non-coplanar unit vectors.
The angle between a, b and b, c is 60º.
a * b = |a| |b| sin(θ) n, where θ is the angle between a and b, and n is the unit vector perpendicular to the plane containing a and b.
b * c = |b| |c| sin(ϕ) n', where ϕ is the angle between b and c, and n' is the unit vector perpendicular to the plane containing b and c.
We need to find 2(p^2 q^2 r^2).
Step 1: Find a * b * c.
a * b * c = (a * b) * c = (|a| |b| sin(θ) n) * c
= |a| |b| sin(θ) (n * c)
= -|a| |b| sin(θ) (c * n)
= -a * (b * c)
= -a * (|b| |c| sin(ϕ) n')
= -|a| |b| |c| sin(θ) sin(ϕ) (n * n')
Step 2: Find the value of p, q, and r.
a * b * c = pa qb rc
=> -|a| |b| |c| sin(θ) sin(ϕ) (n * n') = pa qb rc
=> 2(p^2 q^2 r^2) = (sin(θ) sin(ϕ))^2
Step 3: Substitute the values of sin(θ) and sin(ϕ).
sin(θ) = sin(60º) = √3/2
sin(ϕ) = sin(60º) = √3/2
=> 2(p^2 q^2 r^2) = (√3/2)^2 * (√3/2)^2
= 3/4
Therefore, 2(p^2 q^2 r^2) = 3/4, or p^2 q^2 r^2 = 3/8.
Let a, b, c be non-coplanar unit vectors, making an angle of 60º with each other. If a * b b * c = pa qb rc, then the value of 2(p^2 q^2 r^2) is equal to what?
Solution
Given:
a, b, c are non-coplanar unit vectors.
The angle between a, b and b, c is 60º.
a * b = |a| |b| sin(θ) n, where θ is the angle between a and b, and n is the unit vector perpendicular to the plane containing a and b.
b * c = |b| |c| sin(ϕ) n', where ϕ is the angle between b and c, and n' is the unit vector perpendicular to the plane containing b and c.
We need to find 2(p^2 q^2 r^2).
Step 1: Find a * b * c.
a * b * c = (a * b) * c = (|a| |b| sin(θ) n) * c
= |a| |b| sin(θ) (n * c)
= -|a| |b| sin(θ) (c * n)
= -a * (b * c)
= -a * (|b| |c| sin(ϕ) n')
= -|a| |b| |c| sin(θ) sin(ϕ) (n * n')
Step 2: Find the value of p, q, and r.
a * b * c = pa qb rc
=> -|a| |b| |c| sin(θ) sin(ϕ) (n * n') = pa qb rc
=> 2(p^2 q^2 r^2) = (sin(θ) sin(ϕ))^2
Step 3: Substitute the values of sin(θ) and sin(ϕ).
sin(θ) = sin(60º) = √3/2
sin(ϕ) = sin(60º) = √3/2
=> 2(p^2 q^2 r^2) = (√3/2)^2 * (√3/2)^2
= 3/4
Therefore, 2(p^2 q^2 r^2) = 3/4, or p^2 q^2 r^2 = 3/8.
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?Let a, b, c be non coplanar unit vectors, making an angle of 60º with each other. If a *b+ b* c =pa qb rc , then the value of 2(p^2 q^2 r^2) is equal to * = here is for cross product?
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?Let a, b, c be non coplanar unit vectors, making an angle of 60º with each other. If a *b+ b* c =pa qb rc , then the value of 2(p^2 q^2 r^2) is equal to * = here is for cross product? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about ?Let a, b, c be non coplanar unit vectors, making an angle of 60º with each other. If a *b+ b* c =pa qb rc , then the value of 2(p^2 q^2 r^2) is equal to * = here is for cross product? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ?Let a, b, c be non coplanar unit vectors, making an angle of 60º with each other. If a *b+ b* c =pa qb rc , then the value of 2(p^2 q^2 r^2) is equal to * = here is for cross product?.
?Let a, b, c be non coplanar unit vectors, making an angle of 60º with each other. If a *b+ b* c =pa qb rc , then the value of 2(p^2 q^2 r^2) is equal to * = here is for cross product? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about ?Let a, b, c be non coplanar unit vectors, making an angle of 60º with each other. If a *b+ b* c =pa qb rc , then the value of 2(p^2 q^2 r^2) is equal to * = here is for cross product? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for ?Let a, b, c be non coplanar unit vectors, making an angle of 60º with each other. If a *b+ b* c =pa qb rc , then the value of 2(p^2 q^2 r^2) is equal to * = here is for cross product?.
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?Let a, b, c be non coplanar unit vectors, making an angle of 60º with each other. If a *b+ b* c =pa qb rc , then the value of 2(p^2 q^2 r^2) is equal to * = here is for cross product?, a detailed solution for ?Let a, b, c be non coplanar unit vectors, making an angle of 60º with each other. If a *b+ b* c =pa qb rc , then the value of 2(p^2 q^2 r^2) is equal to * = here is for cross product? has been provided alongside types of ?Let a, b, c be non coplanar unit vectors, making an angle of 60º with each other. If a *b+ b* c =pa qb rc , then the value of 2(p^2 q^2 r^2) is equal to * = here is for cross product? theory, EduRev gives you an
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