What is a scalar? A scalar is a quantity that is fully described by a magnitude only. It is described by just a single number. Some examples of scalar quantities include speed, volume, mass, temperature, power, energy, and time. What is a vector? A vector is a quantity that has both a magnitude and a direction. Vector quantities are important in the study of motion. Some examples of vector quantities include force, velocity, acceleration, displacement, and momentum.
[The military chambers of llanberis, puddled doorway] A displacement is the shortest distance from the initial to the final position of a point P. Thus, it is the length of an IMAGINARY STRAIGHT PATH, typically distinct from the path actually travelled by P. A displacement vector represents the length and direction of this imaginary straight path.
The trajectory of a particle (in red) along a curve inside a vector field. Starting from a, the particle traces the path C along the vector field F. The dot product (green line) of its displacement vector (red arrow) and the field vector (blue arrow) defines an area under a curve, which is equivalent to the path's line integral.
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