NOTES 4.1
Woo Hoo!!!
New Semester!!!
Fresh Start!!!!
Pre-Calculus
Section 6.3 Vectors in the Plane
Name: _____________________________
Some quantities require two numbers because they involve both a magnitude and direction.
Examples:
Quantities that have magnitude and direction are called VECTORS and are represented
geometrically as a directed line segment (similar to a ray from geometry).
Q
Point P is the ______________________.
Point Q is the _____________________.
P
This vector can be written in a couple ways:
Magnitude:
If directed line segments have the same direction and the same magnitude, they are equivalent.
How can you show that
two vectors
are equivalent???
V
U
U
V
Vectors in standard position.
A vector is in standard position if and only if:
V
U
U
V
Component form of a vector:
Magnitude of a vector:
Zero Vector:
Unit Vector:
Scalar Multiplication
An operation where a scalar k is “multiplied” by vector v to produce another vector that is kv.
Scalar multiplication affects the ______________ of the vector but not the ______________
except when the scalar is a _______________.
Vector Addition/Subtraction
Let u= u1 , u2 and v= v 1 , v 2 .
The sum of u and v is the vector:
The difference is the vector:
The difference can also be thought of as u + (-v).
Unit Vectors
You can find a unit vector that has the same direction as a given nonzero vector v using the formula:
Note that u is a scalar multiple of v. The vector u has a magnitude of 1 and the same direction as v.
The vector u is called a unit vector in the direction of v.
The unit vectors 1,0 and 0,1 are called the standard unit vectors and are denoted by:
i = 1,0 and j = 0,1
For example, the vector u = 5,7 is represented as a linear
combination of i and j as follows:
If v = v 1 , v 2 , then the linear combination representation is:
Ex: Given v= 4,2 , find a unit vector in with the same direction as v. Then write both vectors as a
linear combination of vectors i and j.
Ex: Let u be a vector with initial point (2,-5) and terminal point (-1,3). Write as a linear
combination of the standard unit vectors i and j. Then find a unit vector with the same direction
as vector u.
Ex: Let u = -3i + 8j and let v = 2i 7 3j. Find 2u – 3v.