Question

4) (5 points) Given are the magnitudes and orientations (with respect to x-axis) of 3

vectors:
TĀ = 4.5N, 0,= 55
BI = 1.5N, 0,= 145
C = 6.00, 0 = 235
Determine the resultant vector Ř of these 3 vectors through the analytic method
involving components. You have state both the magnitude and the angle with respect to
x-axis.

Answer 1

Expand each vector into their component forms:

`\vec A=(4.5\,\mathrm N)(\cos\theta_A\,\vec\imath+\sin\theta_A\,\vec\jmath)=(2.58\,\vec\imath+3.69\,\vec\jmath)\,\mathrm N`

Similarly,

`\vec B=(-1.23\,\vec\imath+0.860\,\vec\jmath)\,\mathrm N`

`\vec C=(-3.44\,\vec\imath-4.91\,\vec\jmath)\,\mathrm N`

Then assuming the resultant vector
`\vec R` is the sum of these three vectors, we have

`\vec R=\vec A+\vec B+\vec C`

`\vec R=(-2.09\,\vec\imath-0.368\,\vec\jmath)\,\mathrm N`

and so
`\vec R` has magnitude

`\|\vec R\|=√((-2.09)^2+(-0.368)^2)\,\mathrm N\approx2.12\,\mathrm N`

and direction
`\theta_R` such that

`\tan\theta_R=(-0.368)/(-2.09)\implies\theta_R=-170^\circ=190^\circ`