8.4 Vectors

A vector is a quantity that has both magnitude and direction. Vectors in the plane can be represented by arrows. The length of the arrow represents the magnitude of the vector. The arrowhead indicates the direction of the vector.

P Q Initial Point Terminal Point Directed line segment

If a vector v has the same magnitude and the same direction as the directed line segment PQ, then we write v = PQ The magnitude of the directed line segment PQ is the distance from point P to the point Q. The direction of PQ is from P to Q.

The vector v whose magnitude is 0 is called the zero vector, 0. Two vectors v and w are equal, written if they have the same magnitude and direction.

v = w w v

Initial point of v Terminal point of w v + w v w

Vector addition is commutative. Vector addition is associative. v + w = w + v v + (u + w) = (v + u) + w v + 0 = 0 + v =v v + (-v) = 0

Multiplying Vectors by Numbers

v 2v -v

Properties of Scalar Products

Use the vectors illustrated below to graph each expression. v w u

v + w

-w 2v v w 2v and -w

2v 2v-w -w

An algebraic vector v is represented as v = where a and b are real numbers (scalars) called the components of the vector v.

If v = is an algebraic vector with initial point at the origin O and terminal point P = (a, b), then v is called a position vector.

P = (a, b) v = x y O

The scalars a and b are called components of the vector v =.

Theorem Suppose that v is a vector with initial point P 1 =(x 1, y 1 ), not necessarily the origin, and terminal point P 2 =(x 2, y 2 ). If v=P 1 P 2, then v is equal to the position vector

Find the position vector of the vector v=P 1 P 2 if P 1 =(-2, 1) and P 2 =(3,4).

O v = P 1 =(-2, 1) P 2 =(3,4).

Theorem Equality of Vectors Two vectors v and w are equal if and only if their corresponding components are equal. That is,

Let i denote a unit vector whose direction is along the positive x-axis; let j denote a unit vector whose direction is along the positive y-axis. Any vector v = can be written using the unit vectors i and j as follows:

Theorem Unit Vector in Direction of v For any nonzero vector v, the vector is a unit vector that has the same direction as v.

Find a unit vector in the same direction as v = 3i - 5j.