mathmatics, statistics, and computation final

dot product for computing angle between vectors

a*b/IaIIbI=cosθ

a1b1+a2b2+a3b3/ √a1²+a2²+a3² * √b1²+b2²+b3²

finding length of a vector

IaI=√a1²+a2²+a3²

unit vector

1/IaI * the vector

1/IaI*(A1,A2,A3)

vector projection of P2 onto P1

proj p1 P2= P2*P1 / IP1I² * P1

4/5 * (2 1 0)

multiplying it by the actual vector

scalar projection of P2 onto P1

P2*P1/IP1I

only difference between this and vector projection is with scalar you don’t square the denominator which is the vector being projected onto and you dont multiply by the vector

size of a matrix

rows*colomns

what is the rule for matrix multiplication

since the size of a matrix is number of rows times colomns; the number of colomuns in the first matrix must be equal to the number of rows in the second matrix

3×2 matrix by a 2×3 matrix

matrix multiplication

inverse of a matrix

1/detA(d/-c -b/a)

detA=ad-bc

how to check if your inverse calculation is correct

A*inverseA=I

original times inverse should equal the identity matrix

even versus odd function

change the x values to be negative and then:

even if f(-x)=f(x)

odd if f(-x)=-f(x)

neither if it doesnt equal either

whats another word for scalar product and how do you find it

dot product= AxBx+AyBy

when are two vectors orthogonal to each other?

at 90degrees or when cosθ=pi/2=0=U*V

when finding the inverse of f(x)

solve for x but include y when multiplying the fraction and replace f(x) with y then solve

even function

f(-x)=f(x)

odd function

f(-x)=-f(x)

f(x)+2

vertical shift up by 2

f(x-2)

horizontal shift right 2

-f(x)

x-axis reflection

2f(x)

vertical stretch

½ f(x)

vertical stretch

f(2x)

horizontal shrink

when finding domain of f/g numbers under the sqaure root cannot be negative

inequality and make sure it makes sense