A force of \(\text{17}\) \(\text{N}\) in the positive \(x\)-direction acts simultaneously (at the same time) to a force of \(\text{23}\) \(\text{N}\) in the positive \(y\)-direction. Calculate the resultant force.

We draw a rough sketch:

Now we determine the length of the resultant.s

We note that the triangle formed by the two force vectors and the resultant vector is a right-angle triangle. We can thus use the Theorem of Pythagoras to determine the length of the resultant. Let \(R\) represent the length of the resultant vector. Then: \begin{align*} F_x^{2} + F_y^{2} &= R^{2}\ \text{Pythagoras' theorem}\\ (17)^{2} + (23)^{2} &= R^{2}\\ R &= \text{28,6}\text{ N} \end{align*}

To determine the direction of the resultant force, we calculate the angle α between the resultant force vector and the positive \(x\)-axis, by using simple trigonometry: \begin{align*} \tan\alpha &= \frac{\text{opposite side}}{\text{adjacent side}} \\ \tan\alpha &= \frac{\text{23}}{\text{17}} \\ \alpha &= \tan^{-1}(\text{1,353}) \\ \alpha &= \text{53,53}\text{°} \end{align*}

The resultant force is then \(\text{28,6}\) \(\text{N}\) at \(\text{53,53}\)\(\text{°}\) to the positive \(x\)-axis.