Sign up for your FREE 7-day trial.Get instant access to all the benefits of SparkNotes PLUS! Cancel within the first 7 days and you won't be charged. We'll even send you a reminder.
SparkNotes Plus subscription is $4.99/month or $24.99/year as selected above. The free trial period is the first 7 days of your subscription. TO CANCEL YOUR SUBSCRIPTION AND AVOID BEING CHARGED, YOU MUST CANCEL BEFORE THE END OF THE FREE TRIAL PERIOD. You may cancel your subscription on your Subscription and Billing page or contact Customer Support at custserv@bn.com. Your subscription will continue automatically once the free trial period is over. Free trial is available to new customers only.
Step 2 of 4
Choose Your Plan
Step 3 of 4
Add Your Payment Details
Step 4 of 4
Payment Summary
Your Free Trial Starts Now!
For the next 7 days, you'll have access to awesome PLUS stuff like AP English test prep, No Fear Shakespeare translations and audio, a note-taking tool, personalized dashboard, & much more!
Thanks for creating a SparkNotes account! Continue to start your free trial.
Please wait while we process your payment
Your PLUS subscription has expired
We’d love to have you back! Renew your subscription to regain access to all of our exclusive, ad-free study tools.
Did you know you can highlight text to take a note?x
In order to represent physical quantities such as position and momentum in more
than one dimension, we must introduce new mathematical objects called
vectors. Technically speaking, a vector is defined as an element of a
vector space, but since we will only be dealing with very special types of
vector spaces (namely, two- and three-dimensional Euclidean space) we can be
more specific. For our purposes, a vector is either an ordered pair or triplet
of numbers. On a two-dimensional plane, for instance, any point (a, b) is a
vector. Graphically, we often represent such a vector by drawing an arrow from
the origin to the point, with the tip of the arrow resting at the point. The
situation for three-dimensional vectors is very much the same, with an ordered
triplet (a, b, c) being represented by an arrow from the origin to the
corresponding point in three-dimensional space.
The vector (a, b) in the Euclidean plane.
Unlike scalars, which have only a value for magnitude, vectors are often
described as objects that have both magnitude and direction. This can be
seen intuitively from the arrow-like representation of a vector in the plane.
The magnitude of the vector is simply the length of the arrow (i.e. the distance
from the point to the origin), and can be easily computed using the
Pythagorean Theorem. The direction of a
vector in two dimensions can be characterized by a single angle θ(see
); the direction of a vector in three dimensions can be
specified using two angles (usually denoted θ and μ).
While these ideas are perfectly valid in our case (since we're dealing with
vectors in finite-dimensional Euclidean space) it is not a good idea to become
too attached to the notions of "direction" and "magnitude" for vectors. For
instance, in quantum mechanics vectors often come in the
form of functions (for instance, a particle wave function), and in such a case
it doesn't make sense to talk about the "direction" of the vector. We don't
have to worry about these complications for now, though, and in the following
SparkNote we will rely heavily on basic geometric notions when we discuss vector
addition and multiplication.