tilavektorilla

Tilavektori, or position vector, is a vector that locates a point in a space relative to a chosen origin. In three-dimensional Euclidean space it is typically denoted r and can be written in component form as r = (x, y, z) or in unit-vector form as r = x i + y j + z k. The Finnish form tilavektorilla is the instrumental/adessive grammatical form used in sentences, while tilavektori is the noun for the concept itself.

The magnitude of the position vector gives the distance from the origin to the point: |r| = sqrt(x^2 + y^2 + z^2). The direction of r is from the origin toward the point. The position vector embodies the location information of a point in space, independent of how the point is described in other coordinate systems.

In dynamics, the position vector is time-dependent: r = r(t). Its first derivative is the velocity, v = dr/dt, and the second derivative is the acceleration, a = d^2r/dt^2. These relations hold in any inertial frame and form the basis for expressing Newton’s laws in vector form: F = m a.

In different coordinate systems, the position vector still represents the point from the origin, but its component form changes. In Cartesian coordinates it is (x, y, z); in cylindrical coordinates the radial distance ρ is the magnitude of the projection onto the horizontal plane, and in spherical coordinates the radial coordinate r equals |r|.

Applications include computing projections (r · u for a unit vector u), angular momentum L = r × p, and general formulation of motion and forces in physics and engineering.

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