Add the following vectors
1.50m{North 45 degrees West}, 10m{South30degreesEast}, 25Meters[South60mWest]
pls help. explain to me how to solve it
Add the x and y components separately.
x (east) components: -1.5cos45 + 10cos30 -25 cos60
y (north) components: + 1.5 sin45 -10sin30 -25sin60
thanks
To add vectors, you need to break them down into their horizontal (x-axis) and vertical (y-axis) components. Then, you can add the horizontal components together and the vertical components together to obtain the resultant vector.
1. Let's start with the first vector: 1.50m{North 45 degrees West}. To break it down into its x and y components:
- The angle given is measured from the North direction in a counterclockwise direction. Since it is North 45 degrees West, we subtract 45 degrees from 90 degrees (North direction) to get 45 degrees.
- Now, we can find the x component by multiplying the magnitude (1.50m) by the cosine of the angle (45 degrees), which is cos(45 degrees) = 0.707. So, the x component is 1.50m * 0.707 = 1.06m (approximately).
- Similarly, the y component can be found by multiplying the magnitude (1.50m) by the sine of the angle (45 degrees), which is sin(45 degrees) = 0.707. Thus, the y component is 1.50m * 0.707 = 1.06m (approximately).
2. Now let's move on to the second vector: 10m{South 30 degrees East}:
- The given angle is measured from the South direction in a counterclockwise direction. Since it is South 30 degrees East, the angle we consider is 30 degrees.
- To find the x component, we multiply the magnitude (10m) by the cosine of the angle (30 degrees). So, the x component is 10m * cos(30 degrees) = 10m * 0.866 = 8.66m (approximately).
- The y component is obtained by multiplying the magnitude (10m) by the sine of the angle (30 degrees). Thus, the y component is 10m * sin(30 degrees) = 10m * 0.5 = 5m.
3. Lastly, we have the vector 25 meters[South60mWest]:
- Here, the magnitude is 25 meters.
- The angle given is measured from the South direction. Since it is South 60 degrees West, the angle we consider is 150 degrees (180 degrees - 60 degrees).
- The x component can be obtained by multiplying the magnitude (25m) by the cosine of the angle (150 degrees). So, the x component is 25m * cos(150 degrees) = 25m * (-0.866) = -21.65m (approximately).
- The y component is found by multiplying the magnitude (25m) by the sine of the angle (150 degrees). Thus, the y component is 25m * sin(150 degrees) = 25m * (-0.5) = -12.5m.
Now that we have broken down all the vectors into their x and y components, we can add the x-components together and the y-components together. Finally, we can find the magnitude and direction of the resultant vector.
Adding the x-components: 1.06m + 8.66m + (-21.65m) = -12.93m
Adding the y-components: 1.06m + 5m + (-12.5m) = -6.44m
The resultant vector can be found using the components we calculated:
- The magnitude is given by the formula: √(x-component^2 + y-component^2) = √((-12.93m)^2 + (-6.44m)^2) ≈ 14.47m (rounding to two decimal places).
- The direction can be determined using the inverse tangent (arctan) of the y-component divided by the x-component: arctan(-6.44m / -12.93m) ≈ 26.09 degrees (rounding to two decimal places).
Therefore, the sum of the given vectors is approximately 14.47m in magnitude and is directed at an angle of 26.09 degrees (counterclockwise from the positive x-axis).