6. Three vectors are oriented as shown in Figure 3, where |A| = 20.0 units, |B| = 40.0 units,
and |C| = 30.0 units.
a) Find the horizontal and vertical component of the
resultant vector,
b) Magnitude of the resultant vector,
c) The direction of the resultant vector
\45,0
45.0
To find the horizontal and vertical components of the resultant vector, we can use the following trigonometric relationships:
Horizontal Component = |A|cos(thetaA) + |B|cos(thetaB) + |C|cos(thetaC)
Vertical Component = |A|sin(thetaA) + |B|sin(thetaB) + |C|sin(thetaC)
In this case, since all three vectors have the same angle of 45 degrees, we have:
Horizontal Component = |A|cos(45) + |B|cos(45) + |C|cos(45)
= (20.0)(cos(45)) + (40.0)(cos(45)) + (30.0)(cos(45))
= 20.0√2 + 40.0√2 + 30.0√2
= 90.0√2
Vertical Component = |A|sin(45) + |B|sin(45) + |C|sin(45)
= (20.0)(sin(45)) + (40.0)(sin(45)) + (30.0)(sin(45))
= 20.0√2 + 40.0√2 + 30.0√2
= 90.0√2
The magnitude of the resultant vector can be found using the Pythagorean theorem:
Magnitude = sqrt((Horizontal Component)^2 + (Vertical Component)^2)
= sqrt((90.0√2)^2 + (90.0√2)^2)
= sqrt(2×(90.0^2))
= sqrt(2)×90.0
≈ 127.28 units
To find the direction of the resultant vector, we can use the inverse tangent function:
Direction = arctan(Vertical Component / Horizontal Component)
= arctan((90.0√2) / (90.0√2))
= arctan(1)
≈ 45.0 degrees
Therefore, the answers are:
a) The horizontal component of the resultant vector is 90.0√2 units and the vertical component is also 90.0√2 units.
b) The magnitude of the resultant vector is approximately 127.28 units.
c) The direction of the resultant vector is approximately 45.0 degrees.