Given u = 〈-2, 5〉 and v = 〈4, 3〉, find each of the following.
-5v

Key Concepts

Vector Operations

Vector operations involve mathematical manipulations of vectors, which are quantities defined by both magnitude and direction. Common operations include addition, subtraction, and scalar multiplication. In this case, multiplying vector v by a scalar (-5) will scale the vector's magnitude while maintaining its direction, effectively reversing it since the scalar is negative.

Scalar Multiplication

Scalar multiplication is the process of multiplying a vector by a scalar (a real number). This operation alters the magnitude of the vector without changing its direction, unless the scalar is negative, which reverses the direction. For example, if v = 〈4, 3〉, then -5v results in a new vector that is five times longer than v but points in the opposite direction.

Vector Representation

Vectors can be represented in component form, typically as ordered pairs or triples in two or three dimensions. For instance, the vector v = 〈4, 3〉 indicates a movement of 4 units in the x-direction and 3 units in the y-direction. Understanding this representation is crucial for performing operations like scalar multiplication and visualizing the resulting vectors in a coordinate system.