Notes 6.6 – Vectors
Pre-Calc. for AP Prep.
Date: _________
Goal: - Defining vectors and operations with vectors.
Vocabulary:
Vector Quantity – A quantity with ____________ and _________________.
Ex:
25 m.p.h. N30oW
Scalar Quantity – A quantity with just _____________. Ex: $40.
Directed line Segment – A line segment where one endpoint is defined as the
_________________ point and the other is considered the _________________ point.
Magnitude – The size (not considering direction)
Equality – When two vectors have the same size and direction, they are considered equal.
B
v
Notation:
A
Examples: Draw the following vector. Then find its magnitude. (Note the formula and symbol
for magnitude)
1)
v has an initial point of (2, 2)
and terminal point of (5, 6)
2)
u has an initial point of (-2, 6) and
and terminal point of (-5, 2)
3)
u is the vector with initial point (0,0) and terminal point (3, 6). v is the vector with initial
point (-3, -3) and terminal point (0, 3). Show u = v. (Note that we so not have the
flexibility we used to have when we found slope in Algebra I).
Operations with vectors:
Scalar multiplication – When you multiply a vector v by a scalar k, the resulting vector has:
1. A magnitude of _______________________
2. A direction that is the same as v if ___________________ and the opposite of
v if __________________.
Addition of two vectors
v
u
To add u + v , draw the
initial point of v at the
terminal point of u.
Subtraction of two vectors
To subtract u – v , draw the
initial point of -v at the
terminal point of u.
u
v
More Vocabulary:
 The i and j unit vectors
v
So that calculations (rather than drawings)
can be used in vector mathematics, we define
a vector in terms of horizontal and vertical
movement. That is
j
i
We call this representing a vector as a position vector.
Unit Vector:
A vector with the same directions as v, but whose magnitude is 1.
The formuala:
Examples:
4)
Sketch the vector v = -3i + 4j and find its magnitude
5)
Let v be the vector from initial point (3, -1) to terminal point (-2, 5). Write v in terms of i
and j.
If v = 5i + 4j and w = 6i – 9j, find the following:
6)
v+w
7)
v–w
8)
w–v
9)
6v
10)
-3w
11)
Unit vector of
v
Applications:
12)
The wind is blowing at 20 miles per hour in the direction of N30W. Express its velocity
as a vector v in terms of i and j.
13)
Two forces F1 and F2 of magnitude 10 and 30 pounds respectively, act on an object. The
direction of F1 is N20E and the direction of F2 is N65E. Find the magnitude and direction
of the resultant force. Round to the nearest hundredth for the magnitude and the nearest
tenth for the directed angle.
HW: p. 750 #4 – 52 M4, 66, 70, 74
On p. 753 know how to answer 87 – 103