Add the two component vectors from Sample Problem 1 algebraically to erify that they equal the given vector
The two component vectors are:
8.6m(North) and 12m (west)
The given vector is 15m(West 35 degrees North)
To add the two component vectors algebraically, we need to break down the given vector into its horizontal (x-axis) and vertical (y-axis) components.
First, let's determine the horizontal component of the given vector. We know that the vector is 15m(West 35 degrees North), which means it has a magnitude of 15m and points in a direction 35 degrees north of west.
To find the x-component, we need to calculate the horizontal displacement. Given that the vector is pointing west, the horizontal displacement is negative.
Using trigonometry, we can find the x-component as follows:
x = magnitude * cos(angle)
In this case, the magnitude is 15m and the angle is 35 degrees. Therefore:
x = 15m * cos(35)
x ≈ -12.232m (rounded to three decimal places)
The horizontal component of the given vector is approximately -12.232m.
Next, let's determine the vertical component of the given vector. Again, using trigonometry, we can find the y-component as follows:
y = magnitude * sin(angle)
In this case:
y = 15m * sin(35)
y ≈ 8.565m (rounded to three decimal places)
The vertical component of the given vector is approximately 8.565m.
Now, we can add the component vectors algebraically. Given that the two component vectors are 8.6m (North) and 12m (West), we can simply add their magnitudes and directions.
The horizontal component of the result is obtained by subtracting the magnitude of the component vector pointing west from the magnitude of the given vector's x-component:
horizontal component = x-component given vector - x-component component vector
horizontal component = -12.232m - 12m
horizontal component = -24.232m
The vertical component of the result is obtained by adding the magnitudes of the component vectors pointing north and south:
vertical component = y-component given vector + y-component component vector
vertical component = 8.565m + 8.6m
vertical component = 17.165m
Finally, we combine the horizontal and vertical components to form the resultant vector:
Resultant vector = horizontal component + vertical component
Resultant vector = (-24.232m) + 17.165m
Resultant vector ≈ -7.067m (rounded to three decimal places)
Therefore, the algebraic addition of the two component vectors equals approximately -7.067m.