Vectors are mathematical entities characterized by both magnitude and direction. Unlike scalars, which are defined solely by magnitude, vectors represent quantities like displacement, velocity, and force, where direction is essential. Vectors are graphically represented as directed line segments, extending from an initial point to a terminal point, denoted with bold letters or arrows placed above the symbol. Two vectors are deemed equal if they share identical magnitudes and directions, regardless of their initial positions.
Adding vectors involves placing the initial point of one vector at the terminal point of another. Alternatively, if the vectors start from the same point, their sum is the diagonal of the parallelogram formed. Subtracting vectors involves adding the negative of the second vector. Geometrically, the difference forms the opposite diagonal of the parallelogram.
Multiplying a vector by a scalar affects its magnitude, stretching, or shrinking. A negative scalar reverses the vector's direction. For instance, if vector v is multiplied by -2, the resultant vector points in the opposite direction and is twice as long.
Vectors are represented as ordered pairs of their horizontal and vertical components. If a vector v is represented in the plane with initial point A〈x1, y1⟩ and terminal point B〈x2, y2⟩, then the vector v can be written as:
The magnitude of a vector v = 〈a, b⟩ is derived using the Pythagorean theorem:
Vectors can also be expressed as linear combinations of unit vectors. A unit vector is a vector of length one, typically denoted as i for the horizontal component and j for the vertical component. A vector v can be written as:
Here, a and b are the horizontal and vertical components, respectively. These components can also be derived from the vector's magnitude and direction angle θ:
This representation simplifies calculations involving vector addition, subtraction, and scalar multiplication, especially when dealing with vector components in the Cartesian plane.
Vectors possess both magnitude and direction. Magnitude indicates strength, while direction specifies orientation in space. For example, a car's velocity of 50 km/hr east is a vector, defining its speed and direction of motion.
Vectors are represented as directed line segments from an initial point to a terminal point, denoted with bold letters or an arrow on top. A half arrow on top is also commonly used; both conventions are valid.
Two vectors are considered equal when they have the same magnitude, which indicates strength, and the same direction.
Adding vectors involves positioning one vector's initial point at the other's terminal point to form the resultant. Alternatively, if both vectors originate from the same point, their sum is represented by the diagonal of the parallelogram they form, starting from that same point.
Subtracting vectors involves adding the negative of a vector, which flips the vector’s direction. The result is shown by the diagonal of the parallelogram, running from the terminal point of the subtracted vector to the terminal point of the other vector.
In the coordinate plane, vectors are represented as ordered pairs of horizontal and vertical components, typically enclosed in angle brackets.
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