Section 6.6

Vectors

Overview

• A vector is a quantity that has both magnitude and direction.

• In contrast, a scalar is a quantity that has magnitude but no direction.

Vector Representation

• A vector is usually represented by a directed line segment, one that has an initial point and a terminal point.

• Vectors are written using a boldface letter, or an arrow over a single letter:

v

Magnitude

• The magnitude of a vector is its length. Use the formula for the distance between points to find the length of a vector:

• Two vectors are equal if they have the same magnitude and the same direction.

2122

12 yyxxv

Example.

• Given vector v with initial point P(5, -2) and terminal point Q(-3, -4):

1.Sketch v.2.Find the magnitude of v.

Unit Vectors

• A unit vector is a vector with a magnitude of 1.

• Vector i is the unit vector whose initial point is at the origin and whose direction is along the positive x-axis.

• Vector j is the unit vector whose initial point is at the origin and whose direction is along the positive y-axis.

More…

• Vectors in the rectangular coordinate system can be represented in terms of i and j:

• If vector v has initial point at the origin and terminal point (a,b), then

• a is the horizontal component and b is the vertical component, and

jbiav

22 bav

More…

• If the initial point of v is not at the origin, then

jyyixxv

1212

Examples

• Let v be the vector from initial point P(-3, -5) to terminal point Q(3, 4).

1.Sketch the graph.2.Find the magnitude of v.3.Write v in terms of i and j.

Vector Arithmetic in Terms of i and j

• If and and k is a real number then:

jbiav11 jbiaw

22

jkbikavk

jbbiaawv

jbbiaawv

11

2121

2121

Examples

• Let u = 2i – 7j and v = -4i + 8j. Find each of the following vectors, written in terms of i and j.

1.u – v2.7u + 5v3.The magnitude of v – u

Unit Vectors re-visited

• For any nonzero vector v, the vector

is the unit vector that has the same direction as v.

vv

Example

• Find the unit vector that has the same direction as the vector v = 6i + 8j

Writing a vector in terms of its magnitude and direction

jvivv sincos

Example: if vector v has a magnitude ||v|| = 32 and a direction θ = 225°, write v in terms of i and j.

Resultant Forces

• When two vectors are acting simultaneously on an object, the resultant force can be found by:

1. Writing each vector in terms of i and j, then adding the vectors together (parallelogram method).

2. Drawing the vectors from “tip to tail”, then using the Law of Sines and/or the Law of Cosines (tip to tail method) to find the magnitude and direction of the resultant force.

Some Pictures

Examples

• The magnitude and direction of two forces acting on an object are 110 pounds, S61°E, and 120 pounds, N54°E, respectively. Find the magnitude and direction of the resultant force.

• MLP, Problem 15.