Question

Three vectors of magnitudes respectively, 1 unit,2 units and 3 units are directed along the threesides of an equilateral triangle. Then the resultantof the three vectors is of -(A) 1 unit(B) 2 unit(C) 13 units (D) 14 units​

Answer 1

Answer:

in the explanation

Explanation:

Let ABC be the equilateral triangle where the side AB is taken on positive x- axis with A at origin . Take AB as base of the triangle and draw the two sides each making an angle of 60 degree with AB at A and B respectively meeting at the point C. Assume vectors or forces of magnitude 1 ,2 and 3 act respectively along the side AB, BC and CA. Let us resolve these forces along x-axis and y- axis. Denote the sum of the components of these forces by X and Y. ( note carefully the direction of the component vectors with respect to the Positive direction of x and y- axis ). Now;

X = [ - 3 Cos 60° + 1 - 2Cos60°] i = ( - 3/2) i ;

Y = [ - 3 Sin 60° + 2Sin 60°] j = -{ ( 1/2 )sqrt (3) } j . Therefore,

Resultant vector R = - (3/2) i - (sqrt (3)/2 ) j and | R | = R = sqrt [ X^2 + Y^2 ] = sqrt [ 12/4 ] = sqrt(3 ) and it makes an angle θ (say) with x - axis where tan( θ) = ( Y/X ) = [ (-1/2) sqrt (3) /( -3/2 ) ] =1/sqrt (3)=150° as both the components have negative sign .