Vector Arithmetic — Question 4

Question 4

A boat moves at $12$ km/h due east relative to the water while a current flows at $5$ km/h due north. Find the ground velocity, speed, and direction north of east.

Question 4 – Solution

Write every vector in matching component order. Perform scalar multiplication before addition or subtraction, then interpret the resulting components in the context of the problem.

Add the velocity vectors: $\vec v_g=\langle12,0\rangle+\langle0,5\rangle=\langle12,5\rangle$.

The ground speed is $\sqrt{12^2+5^2}=13$ km/h.

The direction satisfies $\tan\theta=5/12$, so $\boxed{\theta=\tan^{-1}(5/12)\approx22.6^\circ}$ north of east.

The result follows from the defining vector formulas used above, and each component, magnitude, or scalar condition has been checked against the information in the question.

Vector equations behave like ordinary algebra provided that every operation is performed component by component. A final component check is often the fastest verification.