Evaluating magnitude of vector

[Raghav Gupta]

Homework Statement

Let a and b be two unit vectors such that | a + b| = √3. If
c = a+ 2b + 3(a x b), then 2|c| is equal to:
√55
√51
√43
√37

Homework Equations

a x b = |a| |b| sinθ n where n is a unit vector
$| a + b | = \sqrt{a^2 + b^2 + 2abcosθ}$

The Attempt at a Solution

Found cosθ = 1/2 and θ = π/3
Then 3 a x b = 3√3/2 n
Now how to find c?
We don't know angle between n and a or n and b

 

[Ray Vickson]

Homework Statement

Let a and b be two unit vectors such that | a + b| = √3. If
c = a+ 2b + 3(a x b), then 2|c| is equal to:
√55
√51
√43
√37

Homework Equations

a x b = |a| |b| sinθ n where n is a unit vector
$| a + b | = \sqrt{a^2 + b^2 + 2abcosθ}$

The Attempt at a Solution

Found cosθ = 1/2 and θ = π/3
Then 3 a x b = 3√3/2 n
Now how to find c?
We don't know angle between n and a or n and b

Yes, you do know the angle. Go back and review the definition and properties of axb.

 

[Raghav Gupta]

Yes, you do know the angle. Go back and review the definition and properties of axb.

I know angle is π/3
What to do next?

 

[Raghav Gupta]

I have also written value of a x b in attempt in post 1

 

[Ray Vickson]

I know angle is π/3
What to do next?

No. The angle between a and b is π/3, but that is not the angle you asked about. You asked about the angle between a and axb or between b and axb.

 

[Raghav Gupta]

Oh that is 90°. But how I will evaluate c?

 

[Ray Vickson]

Oh that is 90°. But how I will evaluate c?

What is preventing you from writing out all the terms of |c|^2 and evaluating them one-by-one? In other words, use the fact that
$$ |\vec{c}|^2 = \vec{c} \cdot \vec{c} \\
= (\vec{a} + 2\vec{b} +3\, \vec{a} \times \vec{b}) \cdot (\vec{a} + 2\vec{b} +3\, \vec{a} \times \vec{b}) $$
and just expand it all out.

 

[Raghav Gupta]

What is preventing you from writing out all the terms of |c|^2 and evaluating them one-by-one?

$|c|^2 = (a+2b)^2 + [3( a X b )]^2 + 2( a+2b)3( a X b ) cos θ$
What is the angle between (a+ 2b) and a x b ?
Between a and a x b it is 90°.

 

[haruspex]

What is the angle between (a+ 2b) and a x b ?

what does the dot product of the two yield?

 

[Raghav Gupta]

It yields 0, so the angle is 90°?

 

[haruspex]

It yields 0, so the angle is 90°?

Yes.

 

[Ray Vickson]

$|c|^2 = (a+2b)^2 + [3( a X b )]^2 + 2( a+2b)3( a X b ) cos θ$
What is the angle between (a+ 2b) and a x b ?
Between a and a x b it is 90°.

You have not "expanded it all out"; you should be getting 6 terms, not just the 3 you have written.

 

[Raghav Gupta]

Got it on solving.
Thanks to both of you.