the method of the coordinates we can find the vector sum of the two vectors given, therefore the answer is:
c = 10.15
a = 70.4º
measured counterclockwise from the positive side of the x-axis
given parameters
to find
The sum of vectors must take into account finding both the magnitude of the vector that is a scalar and its direction.
One of the best methods to perform vector addition is to add their component and then find the resulting vector.
Let's use trigonometry to find the components of vectors a and b, we take the data from the diagram.
Vector a
magnitude m_a = 8.0
angle θ= 45º in the second quadrant
cos 45 = x_a / m_a
sin 45 = y_a / m_a
x_a = m_a cos 45
y_a = m_a sin 45
x_a = 8.0 cos 45 = 5.657
y_a = 8.0 sin 45 = 5.657
as we are in the second quadrant, see diagram
x_a = - 5,657
y_a = 5,627
Vactor b
magnitude m_b = 4.5
angle θ = 60º in the first quadrant
cos 60 = x_b / m_b
sin 60 = y_b / m_b
x_b = m_b cos 60
y_b = m_b sin 60
x_b = 4.5 cos 60 = 2.25
y_b = 4.5 sin 60 = 3.897
Having the components encode the components of the resulting vector
cₓ = x_a + x_b
c_y = y_a + y_b
cₓ = - 5.657 + 2.25 = 3.407
c_y = 5,657 + 3,897 = 9,554
With these values we can find the modulus of the vector using the Pythagoras Theorem
c = [tex]\sqrt{c_x^2 + c_y^2}[/tex]
c = [tex]\sqrt{3.407^2 + 9.554^2}[/tex]
c = 10.14
for the angle we must use the trigonometry relations
tan θ = [tex]\frac{c_y}{c_x}[/tex]
θ = tan⁻¹ [tex]\frac{c_y}{c_x}[/tex]
θ = tan⁻¹ [tex]\frac{9.554}{3.407}[/tex]
θ = 70.4º
In conclusion with the method of coordinates we can find the resulting vector
magnitud c= 10.15
angle θ = 70.4º
measured counterclockwise from the positive side of the x axis
learn more about vector addition here:
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