Vector Calculus

distance between 2 points

√[(x-x1)²+(y-y1)²+(z-z1)²]

equation of a sphere

r²=(x-x1)²+(y-y1)²+(z-z1)²

how to determine collinearity between points A,B,C

1. compute the vectors AB and AC

2. check if AB and AC are multiples of one another

3. If yes, then collinear. Otherwise, no.

parallel vectors definition

they are scalar multiples of each other

how to find a unit vector & its representation

1. find the magnitude of the vector

2. divide all components by the magnitude

strips the vector of its magnitude and only shows us the direction of the vector

how to find segment AB

(b1-a1, b2-a2, b3-a3)

what does the dot product even mean?

how much vectors point in the same direction

1. if dp>0 then they point closely in the same direction (acute)

2. if dp<0 then they point more in the opposite directions (obtuse)

3. if dp = 0 then they are perpendicular (right)

dot product formula

||a||*||b||cos(θ) = a*b = a1b1 + a2b2 + a3b3

what do the direction cosines tell us?

the direction/angle of the vector with respect to the x, y, and z axes

direction cosines formulas

for vector v <a,b,c>

cosα=a/||v||
cosβ=b/||v||

cosγ=c/||v||

cos²α+cos²β+cos²γ=1

how to find unit vectors specific to an angle

1. use the geometric dot product formula and use it as a constraint for VxUx + Vy+Uy

2. make the quadratic equation ux² + uy² = 1

3. solve for ux and uy. 

projection notation

proj_a(b) = projection of vector b onto vector a

proj_b(a) = projection of vector a onto vector b

component of a and component of b, formulas

comp_a(b) = a*b/||a||

comp_b(a) = a*b/||b||

projection of a onto b and projection of b onto a, formulas

proj_a(b) = (a*b/||a||²)a

proj_b(a)=(a*b/||b||²)b

if a X b = 0 then…

a and b are parallel

area of a parallelogram

A = ||a|| * ||b||sinθ = ||a X b||

formula for scalar triple product and what it represents

1. |a*(b X c)|

2. represents the volume of a parallelepiped

how to verify coplanarity using scalar triple product

1. if the stp = 0, then coplanar

2. if the stp =/= 0, then 3d

how to determine if two lines are the same

1. set the equation of one line equal to the initial point of the other.

2. solve for t for each component (x,y,z).

3. if t is the same for all, then yes. otherwise, no

vector, paramteric, and symmetric forms

1. vector: r0 + vt

2. paramteric: x = x0+at, y = y0 + bt, z = z0 + ct

3. symmetric: (x-x0)/a = (y-y0)/b = (z-z0)/c

equation for a plane

normal vector: <a,b,c>

arbitrary vector on the plane: <x-x0, y-y0, z-z0>

equation for a plane: a(x-x0)  + b(y-y0) + c(z-z0) = 0 

this is derived from the dot product!