Vectors Flashcards
What is a vector
Vector is a directed line segment, can be visualised by an arrow
When are 2 vectors parallel/collinear
2 vectors are parallel/collinear if they lie on a straight line. (Zero is considered parallel to all other segments)
WHen are 2 directed line segments equal
2 directed line segments are equal when:
They are collinear
They have same length
They point in same direction
What does scalar multiplication of vector a by L produce
Scalar multiplication of vector a by L produces vector b such that:
B is collinear to a
Norm(b)= mod(L) Norm(a)
A and b have amen direction if L >0, opposite if L<0
What is a unit vector
A unit vector is a vector that has length 1. Â=1/norm(A) * A
What are properties of scalar products
Properties of scalar products are: A.b=b.a A.(b+c)=A.b+A.c L(a.b)=(La).b=a.(Lb) A.a=norm(a)^2
What happens if x.a=x.b
If x.a=x.b, then x.(a-b)=0 so x is orthogonal to a-b.
What happens if x.c=0 for all x
If x.c=0 for all x, then c=0.
X.a=x.b for all x, a=b
What is an orthogonal projection of vector b
Orthogonal projection of vector b = bParallel + bperpendicular
What is vector c=a x b
Vector c =a x b is:
Orthogonal to a and b
Norm(c)= norm(a) * norm(b) * sin(theta)
If a is parallel to b, then a x b =0
What are properties of cross product
Properties of cross product are: A x b = -b x a(anti commutative) A x (b+c) = a x b + a x c L(a x b)= La x b = a x Lb A x a = 0 sin(0)=0
What is the scalar triple product of [a,b,c]
Scalar triple product of [a,b,c] is:
A.(b x c)
= a1(b x c) + a2(b x c) + a3(b x c)
= det matrix with rows of vectors a,b,c
What does it mean if [a,b,c] =0
If [a,b,c] =0, it means a,b,c are coplanar
Means there exists L, U, N such that aL+bU+cN =0
What does cyclic mean STP
Cyclic means [a,b,c] = [b,c,a] = [c,a,b]
