Vectors
ACCELERATED MATH 3
Definitions and Properties (Pg. 453)
 A Vector Quantity is a quantity, such as force,
velocity or displacement, that has both magnitude
(size) and direction.
 A Scalar is a quantity, such as time, speed, or
volume, that has only magnitude, no direction.
 A Vector is a directed line segment that represents a
vector quantity. Symbol: v
 The Tail of a vector is the point where it begins. The
Head of a vector is the point where it ends. An
arrowhead is drawn at the head of a vector.
Definitions and Properties (Pg. 453)
 The Magnitude, or absolute value, of a vector is its
length. Symbol: v . If v  xi  yj , then v 
x2  y 2 .
 A Unit Vector is a vector that is 1 unit long. Vectors
i and j are unit vectors in the x-and y-directions,
respectively. A unit vector u in the direction of a
given vector v is found by u  v .
v
 Two vectors are Equal if they have the same magnitude
and direction. So you can Translate a vector without
changing it, but you can’t rotate or dilate it.
Definitions and Properties (Pg. 453)
 The Opposite of a vector is a vector of the same
length in opposite direction. Symbol: v .
v  xi  yj , starts at the origin
and ends at the point (x,y).
 A Displacement Vector is the difference between an
object’s initial and final positions.
 A Position Vector,
Ex. 1 (Pg. 455)
a) Write these vectors in terms
of their components.
b) Translate w so that its tail is at the head of v . Then
draw the resultant vector r  v  w . Find r
numerically by adding the components of v and w,
and show that the answer agrees with your drawing.
Ex. 1 (Pg. 455)
c) How would you find w  v ?
Why is the answer equivalent to
v w?
d) Find v , w and
explain why v  w
v  w . Based on the graph,
v w.
Ex. 2 (Pg. 456)
a)Draw w as a position vector.
Then translate w so that the tail
is at the head of v . Using the
definition of vector addition draw
v   w . Explain why v  w is equivalent to
v    w .
Ex. 2 (Pg. 456)
b) Draw a displacement vector
from the head of w to the head of
v . Explain why this vector is
equivalent to v  w from part a.
Ex. 2 (Pg. 456)
c) Find v  w numerically from
the coordinates of v and w , and
show that the answer agrees with
your drawings in parts a and b.