T1W8L1: multiplication of vectors

how to take two vectors and multiply them together?? §

• various different ways to do it
• (3i + 2j) x (4i-6j) = (3x4)i + (2*(-6))j
  • turns out only some definitions to be meaningful or useful, and this one isn’t. not at all conventional.
• two methods of multiplying vectors

scalar product §

• mechanical work in physics
  • work = force x displacement ?!??!?!?!? (physics reference IN SPECIALIST???)
  • 200N of force, moves only 5m, therefore work done is 1000Nm
  • how to do it on an angle/??
    • if its 200newtons pulled at an angle of 60 degrees, and it’s moved 5m, the formula is alittle different!!!
    • force in direction = 200N x cos60
  • the scalar product of two vectors $a,b$ is $a\cdot b=|a||b|\cos \theta$
  • SCALAR PRODUCT IS ALWAYS $\vec{a}\cdot \vec{b}$ and NOT $\vec{a}\times \vec{b}$
  • algebraic properties:
    • $a\cdot b=b\cdot a$
    • $a\cdot (\lambda b)=(\lambda a)\cdot b$
    • $a\cdot a=|a{|}^2$
    • $a\cdot (b+c)=a\cdot b+a\cdot c$
    • ALSO:
      • $a\perp b$ if and only if $a\cdot b=0$
  • some other observation:
    • the angle between 2 vectors is always 0, acute, right, obtuse or straight.
    • if it is 0, $\cos \theta =1$, if acute $\cos \theta >0$, if right $\cos \theta =0$, if obtuse $\cos \theta <0$, and if straight $\cos \theta =-1$
    • now replace $\cos \theta$ with $\vec{a}\cdot \vec{b}$!!

  • the miracle component form §

  • $(a\vec{i}+b\vec{j})\cdot (c\vec{i}+d\vec{j})=(a\vec{i})\cdot (c\vec{i})+(a\vec{i})\cdot (d\vec{j})+(b\vec{j})\cdot (c\vec{i})+(b\vec{j})\cdot (d\vec{j})=ac(\vec{i}\cdot \vec{i})+ad(\vec{i}\cdot \vec{j})+bc(\vec{j}\cdot \vec{i})+bd(\vec{j}\cdot \vec{j})=ac+bd$
  • the miracle of component form is hence: $(a\vec{i}+b\vec{j})\cdot (c\vec{i}+d\vec{j})=ac+bd$
  • e.g. if $\vec{u}=5\vec{i}-3\vec{j}$ and $\vec{v}=2\vec{i}+\vec{j}$

vector product §

• next year thingy no need to use