Answer:
The magnitude of the resultant is 55.0 m
The angle of resultant is 18.4° below the x axis
Explanation:
Given that,
Magnitude of vector A = 27 m
Angle = 32° above the x axis
Magnitude of vector B = 35 m
Angle = 35° blow the x axis
We need to calculate the angle between the vectors
Using given angle
[tex]\theta=32-(-55)[/tex]
ptex]\theta=87^{\circ}[/tex]
We need to calculate the sum of vector A and B
Using formula of sum of vector
[tex]C=\sqrt{A^2+B^2+2AB\cos\theta}[/tex]
Put the value into the formula
[tex]C=\sqrt{27^2+35^2+2\times27\times35\cos87}[/tex]
[tex]C=55.0\ m[/tex]
We need to calculate the direction angle
Using formula of angle
[tex]\tan\theta=\dfrac{B\sin\theta}{A+B\cos\theta}[/tex]
Put the value into the formula
[tex]\tan\theta=\dfrac{35\sin87}{27+35\cos87}[/tex]
[tex]\tan\theta=1.21[/tex]
[tex]\theta=\tan^{-1}(1.21)[/tex]
[tex]\theta=50.4[/tex]
We need to calculate the angle of the resultant vector makes with vector A
Using formula of angle
[tex]\phi=32-50.4[/tex]
[tex]\phi=-18.4^{\circ}[/tex]
Negative sign shows that the direction of resultant is below the x axis
Hence, The magnitude of the resultant is 55.0 m
The angle of resultant is 18.4° below the x axis
The magnitude of the sum of the vectors, C, can be obtained using the
parallelogram method.
Reasons:
The given parameters are;
Magnitude of vector A = 27 m
The direction of vector A = 32° above the horizontal
Magnitude of vector B = 35 m
The direction of vector B = 55° above the horizontal
By drawing, using the parallelogram resolution of vectors on MS Word, and
taking extension of the resultant in the x and y-direction, we have;
C = A + B = 42.92·i - 14.48·j
The magnitude of the resultant vector is therefore;
|C| = √((42.92 m)² + (14.48 m)²) ≈ 45.3 m
The direction is given as follows;
[tex]\theta = arctan \left(-\dfrac{14.48}{42.92} \right) \approx -18.6^{\circ}[/tex]
Verifying, we get;
Vector A = 27×cos(32°)·i + 27×sin(32°)·j ≈ 22.9·i + 14.31·j
Vector B = 35×cos(-55°)·i + 35×sin(-55°)·j ≈ 20·08·i - 28.67·j
C = A + B = ( 22.9·i + 14.31·j) + (20·08·i - 28.67·j) = 42.98·i - 14.36·j
Therefore, the vector obtained by sketch is correct approximately.
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