Vector: A mathematical quantity defined by both a magnitude and a direction.
Position Vector: A vector used to specify a location relative to a fixed landmark.
Head and Tail of a Vector: For a vector represented by an arrow, the head is the terminal end (arrowhead), and the tail is the initial point.
Scalar: A quantity that has only magnitude and no direction.
Magnitude of a Vector (geometric): The length of a vector \(\vec{v}\), denoted by \(||\vec{v}||\).
Vector Equality (geometric): Two vectors are equal if and only if they have the same magnitude and direction.
Scalar Multiplication (geometric): The product of a scalar \(k\) and a vector \(\vec{v}\), denoted \(k\vec{v}\), is a vector with the same direction as \(\vec{v}\) if \(k>0\) and the opposite direction if \(k<0\). Its magnitude is \(|k|\) times the magnitude of \(\vec{v}\).
Displacement Vector: A vector that represents the change in position from one point to another.
Resultant Vector: The vector that results from the sum of two or more vectors.
Common Mistakes
Confusing Vectors and Scalars: A vector has both magnitude and direction, while a scalar has only magnitude. The magnitude of a vector, \(||\vec{v}||\), is a scalar.
Adding Magnitudes Instead of Vectors: The magnitude of a sum of vectors, \(||\vec{u} + \vec{v}||\), is generally not equal to the sum of their magnitudes, \(||\vec{u}|| + ||\vec{v}||\). Vector addition must be performed geometrically using the head-to-tail or parallelogram rule.