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Problem

A drone starts at the origin, is intended to move in the direction of $\langle 3, 4 \rangle$, but actually ends at $P = (5, 0)$; find how far it traveled along the intended path and how far it drifted off course.

Step 1: Compute the scalar projection onto the direction vector

Let $D = \langle 3, 4 \rangle$ and $P = \langle 5, 0 \rangle$. The dot product is

$$P \cdot D = 5 \cdot 3 + 0 \cdot 4 = 15.$$

The magnitude of $D$ is

$$|D| = \sqrt{3^2 + 4^2} = 5.$$

So the scalar projection of $P$ onto $D$ is

$$\dfrac{P \cdot D}{|D|} = \dfrac{15}{5} = 3.$$

Step 2: Find the perpendicular drift

The drone's total displacement from the origin to $P$ has length

$$|P| = \sqrt{5^2 + 0^2} = 5.$$

Using the Pythagorean theorem with the along-path distance $3$ and the total distance $5$, the drift is

$$\sqrt{5^2 - 3^2} = \sqrt{16} = 4.$$

Answer

The drone traveled $3$ units along its intended path and drifted $4$ units off course.