Page 1 | © 2012 by Janice L. Epstein 1.1 Introduction to Vectors 
The vector a = AB connecting points A( x1 , y1 ) and B ( x2 , y2 ) is
a = x2 - x1 , y2 - y1
Introduction to Vectors (Section 1.1)
Walk 5 steps. Where are you?
Page 2 | © 2012 by Janice L. Epstein 1.1 Introduction to Vectors 5
EXAMPLE 1
(a) Find the components and magnitude of the vector with initial
point A(-3, 4) and terminal point B(1, 2) .
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5
A vector represents a quantity that has both
magnitude and direction.

(b) Draw AB and the equivalent representation, E , starting at the
origin.
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In 2-d a vector is written a = a = a1 , a2
where a1 and a2 are the components of the vector.
A vector that has the origin as its initial point is
called a position vector.
5
-5
5
The magnitude or length of a vector a = a1 , a2 is given by
a = a12 + a22 = a
The zero vector, 0 = 0,0 , has no length or direction.
(c) Determine the angle that E makes with the positive x-axis.
Page 3 | © 2012 by Janice L. Epstein 1.1 Introduction to Vectors EXAMPLE 2
Find the components of the vector r given that r = 4 and r makes
and angle of 30 with the positive x axis.
Page 4 | © 2012 by Janice L. Epstein 1.1 Introduction to Vectors Given two vectors a = a1 , a2 and b = b1 , b2 , and a scalar c,
a + b = a1 + b1 , a2 + b2
and
ca = ca1 , ca2
EXAMPLE 4
Given that a = 3, -1 and b = 5, -2 find
(a) a + b
A unit vector u is a vector with length 1.
The unit vector in the direction of the positive x-axis is i = 1,0
(b) 2a - 3b + j
The unit vector in the direction of the positive y-axis is j = 0,1
i and j are called the standard basis vectors.
EXAMPLE 3
Write the vectors from Examples 1 and 2 in terms of i and j.
(c) A unit vector in the direction of a
Page 5 | © 2012 by Janice L. Epstein 1.1 Introduction to Vectors Page 6 | © 2012 by Janice L. Epstein 1.1 Introduction to Vectors Many physical quantities such as velocity and force are vector
quantities as they have both direction and magnitude.
EXAMPLE 6
Suppose that a wind is blowing from the direction N45oW at a
speed of 50 km/hr. That means that the direction from which the
wind blows is 45o west of the northerly direction and the
magnitude of the wind velocity is 50 km/hr. A pilot is steering the
plane in the direction N60oE with airspeed (speed in still air) of
250 km/hr. Find the true course and ground speed of the airplane.
EXAMPLE 5
A 50 pound block of stone is
hanging from two cables as shown.
Find the magnitude of T1 and T2
30o
T1
o
45
T2