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Vectors a, b and c are such that a+b+c=0 and |a| =3, |b|=5 and |c|=7 Find the angle between a and b - CBSE (Science) Class 12 - Mathematics

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Question

 

Vectors `veca,vecb and vecc ` are such that `veca+vecb+vecc=0 and |veca| =3,|vecb|=5 and |vecc|=7 ` Find the angle between `veca and vecb`

 

Solution

 

We need to find the angle between `veca and vecb`

Now `(veca+vecb+vecc)^2=(veca+vecb+vecc).(veca+vecb+vecc)`

`(veca+vecb+vecc)^2=(|veca|^2+|vecb|^2+|vecc|^2+2veca vecb+2vecc vecb+2veca vecc)`

`0=(9+25+49+2veca vecb+2vecc vecb+2veca vecc)`

`0=83+2(veca vecb+vecc vecb+veca vecc)`

`-83/2=(veca vecb+vecc vecb+veca vecc)`

`-83/2=(veca vecb+(-veca-vecb) vecb+(-veca-vecb) vecc) (because vecc=-veca-vecb)`

`-83/2=(-|vecb|^2-|veca|^2-vecavecb)`

`-83/2=(-|vecb|^2-|veca|^2-|veca||vecb|cos theta)`

`83/2=(34+15cosθ)`

`⇒83/2−34=15cosθ`

`⇒15/2=15cosθ`

`⇒1/2=cosθ`

`θ=π/3 or (5π)/3`

 
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Solution Vectors a, b and c are such that a+b+c=0 and |a| =3, |b|=5 and |c|=7 Find the angle between a and b Concept: Product of Two Vectors - Scalar (Or Dot) Product of Two Vectors.
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