6. Three vectors are oriented as shown in Figure 3, where 1A| = 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
Figure 3
45.0°
45,0°
To find the horizontal and vertical components of the resultant vector, we can break each vector into its horizontal and vertical components.
Let's denote the horizontal component as Rx and the vertical component as Ry.
For vector A:
Ax = 20.0 units * cos(45°) = 14.14 units (rounded to two decimal places)
Ay = 20.0 units * sin(45°) = 14.14 units (rounded to two decimal places)
For vector B:
Bx = 40.0 units * cos(45°) = 28.28 units (rounded to two decimal places)
By = 40.0 units * sin(45°) = 28.28 units (rounded to two decimal places)
For vector C:
Cx = 30.0 units * cos(0°) = 30.0 units
Cy = 30.0 units * sin(0°) = 0.0 units
a) The horizontal component of the resultant vector is the sum of the horizontal components of A, B, and C:
Rx = Ax + Bx + Cx
= 14.14 units + 28.28 units + 30.0 units
= 72.42 units (rounded to two decimal places)
The vertical component of the resultant vector is the sum of the vertical components of A, B, and C:
Ry = Ay + By + Cy
= 14.14 units + 28.28 units + 0.0 units
= 42.42 units (rounded to two decimal places)
b) The magnitude of the resultant vector, R, can be calculated using the Pythagorean theorem:
|R| = sqrt(Rx^2 + Ry^2)
= sqrt((72.42 units)^2 + (42.42 units)^2)
= sqrt(5247.2164 units)
= 72.43 units (rounded to two decimal places)
c) The direction of the resultant vector, θ, can be calculated using the inverse tangent function:
θ = atan(Ry / Rx)
= atan(42.42 units / 72.42 units)
= 30.47° (rounded to two decimal places)
Therefore, the horizontal component of the resultant vector is 72.42 units, the vertical component is 42.42 units, the magnitude of the resultant vector is 72.43 units, and the direction of the resultant vector is 30.47°.