?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(p2 q2 r2) is equal t?

Question

?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(p2   q2   r2) is equal t?

Answer

Solution

Given

a, b, c are non-coplanar unit vectors making an angle of 60º with each other.

a * b * c = paqbrc

To Find

The value of 2(p2 + q2 + r2)

Explanation

Let’s first calculate a * b and b * c.

a * b = |a| |b| cos(60º) = (1)(1)(1/2) = 1/2

b * c = |b| |c| cos(60º) = (1)(1)(1/2) = 1/2

Now, we can write a * b * c as:

a * b * c = (a * b) * c = (1/2) * c = c/2

Since a, b, and c are unit vectors, we can say that:

|a * b * c| = |c/2| = 1/2

Also, we are given that:

a * b * c = paqbrc

So, we can say that:

|paqbrc| = |c/2| = 1/2

Therefore, |p| |q| |r| = 1

Now, we have to find the value of 2(p2 + q2 + r2).

Let’s first find the value of (p2 + q2 + r2).

We know that |p| |q| |r| = 1.

So, we can write:

p2 q2 r2 = 1

p2 + q2 + r2 = 1/(p2 r2)

Now, we have to find the value of 2/(p2 r2).

We know that:

2/(p2 r2) = (p2 r2 + q2 r2 + p2 q2)/(p2 r2)

2/(p2 r2) = (p2 r2)/(p2 r2) + (q2 r2)/(p2 r2) + (p2 q2)/(p2 r2)

2/(p2 r2) = 1 + (q2 r2)/(p2 r2) + (p2 q2)/(p2 r2)

2/(p2 r2) = 1 + (q/p)2 + (r/p)2

2(p2 + q2 + r2) = 2p2 + 2(q/p)2p2 + 2(r/p)2p2

2(p2 + q2 + r2) = 2p2(1 + (q/p)2 + (r/p)2)

2(p2 + q2 + r2) = 2

Top Courses for JEEView all

Physics for JEE Main & Advanced

Mock Tests for JEE Main and Advanced 2025

Chemistry for JEE Main & Advanced

Mathematics (Maths) for JEE Main & Advanced

Chapter-wise Tests for JEE Main & Advanced

Top Courses for JEE

Top Courses for JEE

Suggested Free Tests

Related Content

JEE Courses

Mock Test Series

Past Year Papers

Additional Courses

Company