How to Find the Magnitude of a Vector
To find the magnitude of a vector, first determine its horizontal and vertical components on their respective number lines around the origin. Next, draw the horizontal and vertical components to plot the point where they intersect. Then draw a line from the origin to that point, creating a vector triangle, which is a right triangle. Finally, use the Pythagorean theorem to calculate the triangle's hypotenuse, which is the same as the vector's magnitude.
Determine the components of the vector. Every vector can be numerically represented in the Cartesian coordinate system with a horizontal (x-axis) and vertical (y-axis) component. It is written as an ordered pair v =< x , y > {\displaystyle v=<x,y>}. For example, the vector above has a horizontal component of 3 and a vertical component of -5, therefore the ordered pair is <3, -5>. Draw a vector triangle. When you draw the horizontal and vertical components, you end up with a right triangle. The magnitude of the vector is equal to the hypotenuse of the triangle so you can use the Pythagorean theorem to calculate it. Rearrange the Pythagorean theorem to calculate the magnitude. The Pythagorean theorem is A + B = C. “A” and “B” are the horizontal and vertical components of the triangle while “C” is the hypotenuse. Since the vector is the hypotenuse you want to solve for “C”. x + y = v v = √(x + y )) Solve for the magnitude. Using the equation above, you can plug in the numbers of the ordered pair of the vector to solve for the magnitude. For example, v = √((3 +(-5) )) v =√(9 + 25) = √34 = 5.831 Don't worry if your answer is not a whole number. Vector magnitudes can be decimals. Determine the components of both points of the vector. Every vector can be numerically represented in the Cartesian coordinate system with a horizontal (x-axis) and vertical (y-axis) component. It is written as an ordered pair v =< x , y > {\displaystyle v=<x,y>}. If you are given a vector that is placed away from the origin of the Cartesian coordinate system, you must define the components of both points of the vector. For example, the vector AB has an ordered pair for point A and point B. Point A has a horizontal component of 5 and a vertical component of 1, so the ordered pair is <5, 1>. Point B has a horizontal component of 1 and a vertical component of 2, so the ordered pair is <1, 2>. Use a modified formula to solve for the magnitude. Because you now have two points you are dealing with, you must subtract the x and y components of each point before you solve using the equation v = √((x 2 -x 1 ) +(y 2 -y 1 ) ). Point A is ordered pair 1 <x 1 , y 1 > and point B is ordered pair 2 <x 2 , y 2 > Solve for the magnitude. Plug in the numbers of your ordered pairs and calculate the magnitude. Using our above example the calculation looks like this: v = √((x 2 -x 1 ) +(y 2 -y 1 ) ) v = √((1-5) +(2-1) ) v = √((-4) +(1) ) v = √(16+1) = √(17) = 4.12 Don't worry if your answer is not a whole number. Vector magnitudes can be decimals. 
Determine the components of the vector. Draw a vector triangle. Rearrange the Pythagorean theorem to calculate the magnitude. Solve for the magnitude. Determine the components of both points of the vector. Use a modified formula to solve for the magnitude. Solve for the magnitude. 