A Euclidean vector is a geometric object that represents both direction and magnitude. It is often depicted as a directed line segment with an initial and terminal point. Vectors are useful for representing physical quantities like movement, and are important in physics for modeling directional and quantitative relationships. Vector addition and subtraction use coordinate geometry, while vector multiplication scales the vector components by a real number factor. Vector concepts can model phenomena like wind effects on aircraft velocity by combining wind and airspeed vectors.

1. Courtney Glavich
MAED 5070

2. A Euclidean Vector is a geometric object
that has both direction and magnitude. A
vector is often labeled with a directed line
segment above it.
Ex. EF -- initial point E, terminal point F

3. Vectors are used to measure movement,
and as such play an important role in
physics since they can represent direction
and magnitude.

4. Generally speaking, the head
of the vector will be at O=
(0,0) , and the head of the
vector will be at (x, y). Here,
the tail of the vector is at O =
(0,0) and the head of the
vector is at A = (2,3).

5. Addition of vectors using coordinates:
If OA = [ ] and OB = [ ] then
X1
Y1
X2
Y2
X1
Y1
X2
Y2
X1 +X2
Y1 +Y2
[ ] + [ ] = [ ] vector addition using coordinates

6. Example from the book: when adding
vectors, you are adding their length to
each other, creating a larger vector.

7. Subtraction of vectors using coordinates:
If OE = [ ] and OF = [ ] then
X1
Y1
X2
Y2
[ ] - [ ] = [ ] vector subtraction using coordinates
X1
Y1
X2
Y2
X1-
Y1-
X2
Y2

8. Example from the book: when
subtracting vectors, you are decreasing
their length from each other creating a
smaller vector.

9. Multiplying vectors by a real number:
If OC = [ ] and aOC =a [ ] then
X
Y
X
Y
[ ]= [ ] real number a times a vector
X
Y
ax
ay

10. Example from the book: when multiplying vectors by a real number, you
are increasing their size by a scale factor.

11. Tailwinds, Headwinds, and Side Winds
To illustrate this principle, consider a plane
flying amidst a tailwind. A tailwind is merely
a wind that approaches the plane from
behind, thus increasing its resulting velocity.
If the plane is traveling at a velocity of 100
km/hr with respect to the air, and if the wind
velocity is 25 km/hr, then what is the velocity
of the plane relative to an observer on the
ground below?

12. Addition of vectors
example, with 100
and 25 being your y
coordinates.

13. consider a plane
traveling with a
velocity of 100
km/hr, South
that encounters
a side wind of
25 km/hr,

14. (100 km/hr)2 + (25 km/hr)2 = R210 000 km2/hr2 + 625 km2/hr2 = R2
10 625 km2/hr2 = R2
SQRT(10 625 km2/hr2) = R
103.1 km/hr = R
Addition of vectors example
using PythagoreanTheorem.

15. Beem, J. K. (2006). Geometry connections: mathematics
for middle school teachers. Upper Saddle River, N.J.:
Pearson Prentice Hall.
Euclidean vector. (n.d.). Princeton University. Retrieved
June 12, 2014, from
http://www.princeton.edu/~achaney/tmve/wiki100k/d
ocs/Euclidean_vector.html
RelativeVelocity and Riverboat Problems. (n.d.).
RelativeVelocity and Riverboat Problems. Retrieved
June 12, 2014, from
http://www.physicsclassroom.com/class/vectors/Less
on-1/Relative-Velocity-and-Riverboat-Problems