In Exercises 5–12, sketch each vector as a position vector and find its magnitude. v = -4i

Key Concepts

Position Vectors

A position vector represents a point in space relative to an origin. In a Cartesian coordinate system, it is expressed in terms of its components along the axes, such as 'v = -4i', which indicates a vector pointing 4 units in the negative x-direction. Understanding position vectors is essential for visualizing and manipulating vectors in two or three dimensions.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the Pythagorean theorem. For a vector expressed in component form, like 'v = -4i', the magnitude is found by taking the square root of the sum of the squares of its components. In this case, the magnitude is |v| = √((-4)²) = 4, indicating the distance from the origin to the point represented by the vector.

Unit Vectors

A unit vector is a vector with a magnitude of one, used to indicate direction without regard to length. It is often derived from a given vector by dividing the vector by its magnitude. Understanding unit vectors is important for normalizing vectors and for applications in physics and engineering, where direction is crucial.