Vectors Flashcards
What are the properties of the norm?
- |⃗v|≥ 0for all⃗ v∈R
- |k⃗v| = |k||⃗v| for all ⃗v ∈ R and all k ∈ R.
-If⃗v̸=⃗0,then the unit vector that points in the same direction as ⃗v is ⃗v .
|⃗v| - For all⃗u,⃗v∈R
What is the triangle inequality
⃗u + ⃗v | ≤ | ⃗u | + | ⃗v | .
What are the formulas for scalar product?
1)
⃗u · ⃗v = |⃗u||⃗v| cos θ
2)
⃗u · ⃗v = u 1 v 1 + u 2 v 2 + u 3 v 3 .
What are the properties of the scalar product
1)
⃗u · ⃗v = ⃗v · ⃗u (commutative law).
2)
⃗u·(⃗v+w⃗)=⃗u·⃗v+⃗u·w⃗ (distributive law).
3)
k(⃗u ·⃗v) = (k⃗u) ·⃗v = ⃗u · (k⃗v).
4)
if a vector does not equal 0, the dot product with itself cannot be less than or equal to 0
What is an orthogonal vector
If two non-zero vectors are at right angles to each other then they are said to be orthogonal or perpendicular.
What makes two vectors orthogonal
Their dot products is 0
What is the vector dot cross product
U*W =
modulus of the matrix
( I, j, k)
(u1,u2,u3)
(w1,w2,w3)
Let ⃗u and w⃗ be vectors in R. Then ⃗u × w⃗ is orthogonal to both ⃗u and w⃗ .
Let ⃗u and w⃗ be vectors in R. Then ⃗u × w⃗ is orthogonal to both ⃗u and w⃗ .
What is modulus (u*w)^2
(mod(U)^2 * mod(w)^2) - (u . w)^2
=
mod(u)^2 * mod(w)^2 * sin^2(theta)
where theta is the angle between u and w
What Is the scalar triple product?
Let u, v and w all be 3d vectors
The scalar triple product is given as
u . (v*w)
when are three vectors co-planar
when the scalar triple product is 0
What is the volume of a parallelopied given by
u . (v*w)
