Question

Two vectors whose magnitudes are in the ratio 1:2
gives resultant of magnitude 30. If angle between
these two vectors is120 then the magnitude of two
vectors respectively will be​

Answer 1

$\Huge{\underline{\underline{\mathfrak{Answer \colon}}}}$

Let x be a common proportion of the two vectors

Given

• Vectors are in the ratio 1:2

The vectors would be "x" and "2x"

• Angle between them is 120,∅ = 120°

• Resultant Vector,R = 30

Law of Cosines

$\displaystyle{\sf{ | \vec{R}| = \sqrt{a {}^{2} + b {}^{2} + 2ab.cos \theta} }}$

Putting the values,we get:

$\sf{30 = \sqrt{x {}^{2} + 4x {}^{2} + 2x {}^{2}.cos120} }$

Now,

cos120 = cos(180 - 60)

Since (180 - ∅) lies in second quadrant,cosine function is negative

» cos120 = - cos60

» cos120 = - 1/2

Now,

$\implies \: \sf{30 = \sqrt{5x {}^{2} - 4x {}^{2}. \frac{1}{2} } } \\ \\ \implies \: \sf{30 = \sqrt{5x {}^{2} - x {}^{2} } } \\ \\ \implies \sf{30 = \sqrt{3x {}^{2} } \implies \: 30 = \sqrt{3}x} \\ \\ \huge{\implies \sf{x = 10 \sqrt{3}}}$

Thus,the magnitudes of the vectors would be 10√3 and 20√3 respectively