You are given three displacement vectors: with magnitude 5.0 m in the direction of 60o below the negative x-axis, vector, has an x-component of +2.5 m and a y-component of +6.0 m and vector has magnitude of 2.5 m in the negative y-axis.
A = 5m[240o]
B = 2.5+6i
C = -2.5i
X = 5*cos240 + 2.5 = 0
Y = 5*sin240 + 6 - 2.5 = -0.83
D = 0-0.83i = 0.83m[-90o].
To find the resultant displacement vector, you need to add the three given displacement vectors together.
Let's start with vector 1, which has a magnitude of 5.0 m and is in the direction of 60 degrees below the negative x-axis. To represent this vector in Cartesian coordinates, we need to find its x-component and y-component.
The x-component can be found by multiplying the magnitude (5.0 m) by the cosine of the angle (60 degrees):
x₁ = 5.0 m * cos(60°) = 5.0 m * 0.5 = 2.5 m
The y-component can be found by multiplying the magnitude (5.0 m) by the sine of the angle (60 degrees):
y₁ = 5.0 m * sin(60°) = 5.0 m * √3/2 ≈ 2.5√3 m ≈ 4.33 m
So, vector 1 can be represented as (2.5 m, -4.33 m).
Now, let's move on to vector 2, which has an x-component of +2.5 m and a y-component of +6.0 m. This vector is already given in Cartesian coordinates, so no conversion is needed.
Vector 2 can be represented as (2.5 m, 6.0 m).
Lastly, vector 3 has a magnitude of 2.5 m in the negative y-axis. Since it lies entirely in the y-axis, its x-component is 0.
Vector 3 can be represented as (0 m, -2.5 m).
To find the resultant vector, you simply add the x-components and y-components of the three vectors:
x_res = x₁ + x₂ + x₃ = 2.5 m + 2.5 m + 0 m = 5.0 m
y_res = y₁ + y₂ + y₃ = -4.33 m + 6.0 m + (-2.5 m) = -0.83 m
Therefore, the resultant displacement vector is (5.0 m, -0.83 m).