T1W7L2: vector in component form

#ae#specialist#vectors

the story so far (like the band) §

• any 2d vector can be resolved into a sum of horizontal and vertical components, written as ai + bj where i and j are unit vectors.
• i is le horizontal vector | magnitude of 1
• j is le vertical vector | magnitude of 1
• if a horizontal vector has a magnitude of 4, then it will be 4i (what?!)
• pretend theres a vertical vector with magnitude of 6, then it will be 6j?!
  • such 4i + 6j = the vector
• that is essentially component form
• ai + bj can be written as <a, b> or $\left(\genfrac{}{}{0ex}{}{a}{b}\right)$
• the miracle
  • vectors in component form can be added by adding their components and simplifying (WHAT?!): $ai+bj+ci+dj=(a+c)i+(b+d)j$
  • incredible stuff
• vectors must undergo conversion
• questions may need your answer in magnitude and direction form, shown as followed:
  • magnitude and direction -> component form -> magnitude and direction.
  • therefore it is an important skill to convert

conversion §

• write the following vector in component form vector
• use trig
• u = ai + bj
• cos(4) = a/5
  • a = 5cos40
• sin(40) = b/5
  • b = 5sin40
• u = 5cos40i + 5sin40j
  • = 3.83i + 3.21j
• a vector v can be written in component form as $v=|v|\cos \theta i+|v|\sin \theta j$
• where 0 is the angle of the vector measured anti-clockwise from east $\vec{V}=|\vec{V}|\cos \theta i+|\vec{v}|\sin \theta j$
• put negative sign on the corresponding thingy if it is not facing up and not facing right.

idk what to title this something something equilibrium §

• find $p$ and $\theta$ if the forces below are in equilibrium.
• the sum of the forces is the zero vector $0i+0j$
• vector equation
  • $25i-50j+ai+bj=0i+0j$
  • $25i+ai+0i$ and $-50j+bj=0j$
  • $25+a=0$ and $-50+b=0$
  • $a=-25$ $b=50$