Calc 3 Chapter 11

(Set notation example): define ∈, ℝ², :

∈ “in/is an element of”

ℝ² “in a 2D plane/all ordered pairs of all real numbers”

: “such that” (can also use |)

easy factor method with an A and B value to get c & what does it factor to

c = (a/2)²
[ x - (a/2) ]²

recall that midpt formula given xyz coordinate is

(.5(x-x₁), .5(y-y₁), .5(z-z₁))

tip to tail addition

parallelogram rule for addition

means you can shift vectors in space

unit vectors î ĵ k̂ & equations such as 5i-3j

i = (1,0,0); j = (0,1,0), k = (0,0,1)

equations can be written like the one above. For the example above,

(5,0,0) (0,-3,0) (0,0,0)

*last is 0,0,0 bc there is no k in the equation

The length (norm) of a vector u^→ = (a,b,c) in the element of a 3D plane is ||u^→|| =

aka this formula shows u how to get the length of a vector

√ ( a² + b² + c² )

the length of a unit vector is always =

1

given that P and Q are points,

(PQ)^→ =

(PQ)^→ = Q - P

note that it is backwards

normalizing a vector formula (scaling it into a unit vector so that it has a length of 1)

scalar dotted w/ a vector

|| c • (u^→) || =

|c| • || (u^→) || for any c ∈ ℝ

triangle inequality of vectors (norm of vectors u + v…)

|| (u^→) + (v^→) || ≤

|| (u^→) || + || (v^→) ||

note the less than or equal to sign

dot product of vectors a and b (non geometric defn) & what does it get you?

a = (a₁ , a₂)

b = (b₁, b₂)

(a^→) • (b^→) =

a₁*b₁ + a₂*b₂

this value shows how much the two vectors point in the same direction
just multiply each x by the other x, each y by the other y, …, add the result of each

dot product (geometric defn)

(a^→) • (b^→) =

||(a^→)|| * ||(b^→)|| * cosθ

note: 0 ≤ θ ≤ π is ALWAYS assumed

Two vectors (a and b) are perpendicular (orthogonal) if the angle between them is π/2 IFF (a^→) • (b^→) = 0

Two vectors (a and b) are perpendicular (orthogonal) if the angle between them is π/2 IFF (a^→) • (b^→) = 0

what does the output from a dot product tell you about two vectors? (+, 0, -)

pos # means angle between the vectors is <90 (they point in same general direction)

0 means the vectors are perpendicular

neg # means the angle between the vectors is >90 (they point in different directions)

projections: how to think of it + what 3 things can u gather from the picture

“if we drop vector a on top of vector b, how much is pointing in the same direction?”

projections are only dependent on direction of vector not length

projection of vector a onto vector b formula

pr_b^→a^→ =

note that the divisor has the norm squared of vector b. vector a is the one being projected onto vector b

how vector addition relates to projections (formula)

given that they are all vectors,
v = pr_vxv + pr_vyv

norm of cross product given vectors a and b

|| a^→ × b^→ || =

|| a^→ || * || b^→ || * sinθ

a^→ × b^→ =

-b^→ × a^→

a^→ × a^→ =

0 for all a^→

determinants for cross product

ad-bc

multiply the diagonals then subtract

cross product: visual representation and important to note that cross product only works in ____ space

a^→ × b^→ is a vector perpendicular to a^→ and b^→

cross product only works in 3D space

right hand rule: what is it and what is it used for

gives you direction of a vector cross product

cross product done mathematically

• note the -j

• note how when you move to the next column, you only take values from all the other columns

area of a parallelogram

area = || a^→ × b^→ ||

volume of a parallelepiped

vector a dotted with the result of crossing vectors b and c

volume = | a^→ • (b^→ × c^→) |

• order of vectors a, b, and c doesn’t matter