6. Three vectors are oriented as shown in Figure 3, where |A| = 20.0 units, |B| = 40.0 units,

Question

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

Answer 1

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.