مقدمة في المتجهات

A vector quantity requires both magnitude and direction to be fully described, whereas a scalar has only magnitude. For example, force is a vector because you must specify both how much force and in which direction. Mass and speed are scalars because they need only a number and unit. This distinction is essential for accurately representing physical phenomena in physics.

Geometrically, vectors are drawn as arrows where the arrow's length represents magnitude and the arrowhead indicates direction. Symbolically, vectors are denoted by italicized letters with an arrow above them, such as vector DAB. The magnitude is shown using the same symbol between vertical bars. This dual representation allows vectors to be analyzed both visually and mathematically.

Parallel vectors point in the same direction, while antiparallel vectors point in exactly opposite directions, 180 degrees apart. Two vectors are equal only if they have identical magnitude and direction. The negative of a vector is obtained by reversing its direction, creating an antiparallel vector with the same magnitude but opposite orientation.

Displacement represents a change in position from one point to another and requires specifying both distance and direction. For instance, telling someone a fishing hole is 6 kilometers away is incomplete without stating the direction, such as northeast. An arrow from starting point A to ending point B visually represents displacement, making direction essential to fully describe the quantity.

Orthogonal vectors have directions perpendicular to each other, forming a 90-degree angle, unlike parallel vectors which point in the same direction. While parallel vectors maintain identical orientation, orthogonal vectors are completely independent in their directional components. Understanding these relationships is fundamental for vector operations and coordinate system analysis.

Two vectors are equal if and only if they have equal magnitudes and the same direction. The position of the vector in space does not matter; only the magnitude and direction determine equality. This definition allows vectors to be translated without changing their identity, which is crucial for vector algebra and physics applications.

Common vector quantities include displacement, velocity, position, force, and torque. Force demonstrates the vector concept well: opening a door requires applying force in a specific direction. Weight, the gravitational force on an object, is also a vector. These examples show why direction is essential alongside magnitude for complete physical description.