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Introduction to vectors

vector notation and operations
by

David Strasburger

on 11 October 2016

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Transcript of Introduction to vectors

Vectors
are quantities with magnitude and direction
how do we indicate that a quantity is vector-valued?
how is a vector specified?
direction-magnitude form
Vector Operations:
addition
subtraction
magnitude
multiplication by scalar
dot product
cross product
"5, 53° north of east"
components, ordered pair, unit-vector form
how long is this vector?
direction-magnitude form
"5, 53° north of east"
components, ordered pair, unit-vector form
Notation:
Vector
Formalism
Representation:
http://drifters.doe.gov/track-a-yoto/track-a-drifter.html
http://en.wikipedia.org/wiki/File:3D_Spherical_2.svg
if vectors a and b were placed tip-to-tail, what vector would go from the tail of the first to the tip of the second?
graphical interpretation
graphical interpretation
magnitude
addition
subtraction
multiplication by scalar
dot product
cross product
direction-magnitude form
questions
is addition commutative? can you prove it?

is it associative? can you prove it?
¿
questions
What happens if the scalar is negative?

Is multiplication by a scalar associative?

how can you use multiplication by scalar to create a unit vector?

such as r-hat in gravitation or Coulomb
¿
by components
if vectors a and b were placed tail-to-tail, what vector would go from the tip of the first to the tip of the second?
graphical interpretation
direction-magnitude form
by components
questions
is subtraction commutative? can you prove it?


¿
-
-
questions
¿
How much do these two vectors point in the same direction?
graphical interpretation
direction-magnitude form
by components
graphical interpretation
direction-magnitude form
by components
graphical interpretation
direction-magnitude form
by components
also called "the scalar product"
Is the dot product commutative?

What does it mean if the dot product is negative?

Can the dot product be zero if both vectors have nonzero magnitudes?

How can you use the dot product to find the included angle?
How much are these two vectors perpendicular?
also called "the vector product"
magnitude:
direction:
by right-hand rule
questions
¿
Is the cross product commutative?

Can the cross product of nonzero vectors be zero?
Full transcript