A vector is a mathematical quantity with both a magnitude and direction. Here is how you find the magnitude of the component vectors. For a two-dimensional vector you need the vectors magnitude denoted by m and its angle denoted by theta. Now you are looking for the magnitude of the x-axis component vector denoted by x and the magnitude of the y-axis component vector denoted by y. The result is that for the magnitude of the x-component vector we have x equals m times cosine of theta. And for the magnitude of the y-component vector we have y equals m times the sine of theta. This is a vector with a magnitude of 200 in an angle of 25 degrees. Finding the component vector magnitudes of this vector results in m equals 200 and theta equals 25 degrees. Such that x equals 200 times the cosine of 25 degrees for 181.26 and y equals 200 times the sine of 25 degrees for a 4.52. These component vectors run along the x and y-axis with magnitude of 181.26 and 84.52 Here is another vector with a magnitude of 150 in an angle of 55 degrees. Finding the component vectors of this vector results in m equals 150 and theta equals 55 degrees. Where x equals 150 times the cosine of 65 degrees which equals 63.39 where y equals 150 times the sine of 65 degrees equals 135.95. These component vectors run along the x and y-axis with magnitude of 63.39 and 135.95 respectively. Now vectors do not add an straight forward manner because you have to consider both their direction and magnitude. Now you first need to find the component vectors of the vectors to be added. In this case we have m1 with component vectors x1 and y1 and m2 with component vectors x2 and y2. This results in a simple addition of magnitude such that x equals x1 plus x2 and y equals y1 plus y2. To find the angle of the new vector denoted as theta we use theta equals the arc tangent of y over x. Now to get the magnitude of the new vector denoted as m we use m equals the square root of x squared plus y squared. Now we will pick out two earlier vectors and add them together. This is you get the component vectors of your vectors we have in this case and one has component vectors of 181.26 and 84.52 and then two has component vectors of 63.39 and 135.95. This results in x equals 181.26 plus 63.39 equals 24.65 and y equals 84.52 plus 135.95 equals 220.47. This results in component vectors for a new vector that run along the x and y axis with magnitudes of 244.65 and 220.47. This is the second one. Now the angle of the new vector is theta equals the arc tangent of 220.47 divided by 244.65 and the magnitude of the new vector is m equals 244.65 squared plus 220.47 squared equals 329.34. Thus we have successfully added the blue and green vectors to produce the red vector.