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 component vectors, which are perpendicular (horizontal and vertical) to each other.
Given Vector: 15m (West 35 degrees North)
We can use trigonometry to determine the magnitudes of the horizontal and vertical components.
Horizontal Component:
Using the cosine function, we can find the magnitude of the horizontal component.
cos(35 degrees) = adjacent/hypotenuse
adjacent = cos(35 degrees) * 15m
adjacent = 0.819 * 15m
adjacent = 12.285m (approximately)
Vertical Component:
Using the sine function, we can find the magnitude of the vertical component.
sin(35 degrees) = opposite/hypotenuse
opposite = sin(35 degrees) * 15m
opposite = 0.574 * 15m
opposite = 8.610m (approximately)
Now that we have the component vectors, we can add them algebraically:
Horizontal Component: 12m (West)
Vertical Component: 8.6m (North)
To add them algebraically, we can simply add the magnitudes and indicate the directions.
Adding the magnitudes:
12m + 8.6m = 20.6m
Direction:
Since the horizontal component is pointing west and the vertical component is pointing north, we need to determine the resulting direction based on the given directions.
The resulting direction can be found using the tangent function:
tan(θ) = opposite/adjacent
tan(θ) = 8.6m/12m
θ = tan^(-1)(8.6m/12m)
θ ≈ 36.87 degrees
Therefore, the resulting vector is 20.6m (approximately) in the direction 36.87 degrees North of West.
By adding the two component vectors, we have calculated the resulting vector to be approximately 20.6m in the direction 36.87 degrees North of West, which verifies that it equals the given vector.