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Q: What is a vector?
short answer: physical quantities that have a magnitude as well as a direction.
eg.
r, v, a in vector
exteded definition (answer): an element of a vector space.
Q: What is a vector space?
answer: a set whose elements are vectors that obey a certain set of rules.
1. If
2. If
3.
4.
5.
6.
7.
8.
9.
10.
eg 0.
eg 1.
cf.
eg 2. V:
eg 3. Laplace equation (→ normal modes)
<basis> a set of linearly independent vectors that spans V. say
<inner product>
a mapping that maps two vectors to a scalar. < vi,vj >1) < au1 + bu2,v > = a < u1,v > + b < u2,v > (a,b are constants)
2)
3)
eg 1. dot product
eg 2.
Dot product
Cross product
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