In physics, quantities are classified as either scalars or vectors. What property or properties completely describe a vector quantity?

Question

If \( \sqrt{\frac{y}{x}} + 4\sqrt{\frac{x}{y}} = 4 \), then \( \frac{dy}{dx} \):

Answer 1

Consider displacement and distance side by side to see why a vector needs both magnitude and direction.

If a person walks 3 kilometre east and then 4 kilometre north, the total distance walked, a scalar, is simply 3 plus 4, which is 7 kilometre.
However the displacement, a vector, is the straight line from the starting point to the final point, which by the Pythagoras rule works out to 5 kilometre, but stating just 5 kilometre is still not the full answer.
To fully describe the displacement we must also state its direction, which here points at some angle north of east, roughly 53 degree from the east direction.
Only when both the 5 kilometre magnitude and this specific direction are given together is the displacement vector completely and unambiguously described.
This is true of every vector quantity, including velocity, acceleration, momentum, and force, all of which need a magnitude and a direction to be fully specified.

Answer 2

Consider displacement and distance side by side to see why a vector needs both magnitude and direction.

If a person walks 3 kilometre east and then 4 kilometre north, the total distance walked, a scalar, is simply 3 plus 4, which is 7 kilometre.
However the displacement, a vector, is the straight line from the starting point to the final point, which by the Pythagoras rule works out to 5 kilometre, but stating just 5 kilometre is still not the full answer.
To fully describe the displacement we must also state its direction, which here points at some angle north of east, roughly 53 degree from the east direction.
Only when both the 5 kilometre magnitude and this specific direction are given together is the displacement vector completely and unambiguously described.
This is true of every vector quantity, including velocity, acceleration, momentum, and force, all of which need a magnitude and a direction to be fully specified.

Show Hint