When you know the x- and y-components of a vector, you know two sides of the vector triangle. The magnitude is the hypotenuse, which you can calculate using the Pythagorean theorem. This theorem states that the square of the hypotenuse is the sum of the squares of the other two sides of the triangle.

Using the vector triangle in the diagram, the Pythagorean theorem says F^{2} = F_{x}^{2} + F_{y}^{2}. If you take the square root of both sides of this equation, then you have solved for the magnitude F in terms of the components, as given in equation (6.3).

(6.3)

$$F=\sqrt{{F}_{x}^{2}+{F}_{y}^{2}}$$

F

=

magnitude of the force (N)

F_{x}

=

x-component of force (N)

F_{y}

=

y-component of force (N)

Magnitude of a vector

Finding the angle

To find the angle of a vector from its components you can use the tangent function in reverse. The inverse tangent is a function that gives you back the angle if you know the ratio of the sides of a right triangle. Mathematically, the inverse tangent of a number n is written as tan^{−1}n. When the number is the ratio of the x- and y-components of a vector, the inverse tangent gives you the angle the vector makes with the x-axis, as given in equation (6.4).

For example, suppose you have a vector $\overrightarrow{F}$ = (+6,+8) N and wish to know the angle this force makes with the x-axis. Applying the inverse tangent results in an angle of +36.9°. When using a calculator, carefully check whether it is set to degree or radian mode. One radian (rad) is about 57.3°, so a 30° angle is the same as 0.52 rad.

Three-dimensional vectors

For any vector in (x, y, z) form, you can find the magnitude using the three-dimensional version of the Pythagorean theorem. The square of the magnitude is the sum of the squares of all three coordinate values. That tells us that the magnitude of force (F_{x}, F_{z}, F_{z}) is the square root of the sum of the squares: F_{x}^{2} + F_{y}^{2} + F_{z}^{2}.

Angles in three dimensions

Angles in three dimensions are trickier because a three-dimensional vector makes a different angle with each of the three coordinate axes. You can still use the inverse tangent to find angles; the geometry is more complicated, however, since both y- and z-components affect how far a vector tilts away from the x-axis and therefore affect the angle.