Chapter 6: Introduction to Vectors
*Given a=(-1,-2,1) and b=(3,-1,1), determine if c=(-9,-4,1) lies on π formed by a and b
*c lies on lies on π formed by a and b iff c=ma+nb m,n∈R
Spanning sets (important points)
-any pair of non-zero, non-collinear vectors will span R² -any pair of non-zero, non-collinear vectors will span a plane in R³ but not all planes (parallel planes don't span each other)
*Linear combinations of vectors
-for non-collinear vectors, u and v, a linear combination is: au+bv a,b∈R -au+bv is simply the diagonal of the geometric vector addition -all vectors in R² can be written as a linear combination of i and j so the set of {i,j} forms a spanning set for R² →there are infinite spanning sets in R²
*Adding zero vectors
0+a=a
Other laws of vector addition and scalar multiplication
1. Adding zero vectors 2. Associate law of scalars 3. Distributive law for scalars
Properties of vectors
1. Commutative property of addition 2. Associate property of addition 3. Distributive property of addition
Two different laws of vector addition
1.Parallelogram addition 2.Triangle addition
*Algebraic vector vs geometric vector
Algebraic vectors are tied to a coordinate system (x-y axis)
*Vector components in R²
Any vector, v, in R² can be broken into vector components vx and vy which will be parallel to i and j respectively and perpendicular to each other
Vector notation
Arrow where the length is proportional to the magnitude and the arrowhead yields direction
*Collinear
Fall on the same line (direction doesn't matter) -all collinear lines are parallel but not all parallel lines are collinear
Vector
Has a magnitude (#) and a direction (up/down/left/right/NESW) -EX: 7m[W]
*Associate property of addition
PS=r=a+b+c=(a+b)+c=a+(b+c) PS=r=PQ+QR+RS+PQ+QS=PR+RS
*Sine law
Requires one complete ratio if the side length and its corresponding angle
*True bearing
Starts at north=0° and moves clockwise
*Parallelogram rule for vector addition
Tail to tail addition
if v=100km/h[E], describe the following vectors a) 2v b) 1/2v c) -3v
a) 200km/h[E] b) 50km/h[E] c) 300km/h[W]
Write the following in terms of vectors: A cyclist is travelling 20km/h[N] a) she reverses b) she halves her velocity c) she turns around and doubles her velocity
a) 20km/h[S] b) 10km/h[N] c) 40km/h[S]
*When given two vectors a and b, how can you: a) maximize a resultant vector (r) b) minimize r
a) parallel and same direction b) parallel and opposite direction
*Commutative property of additon
b+a=a+b=r AB+BC=AD+DC=AC
Multiplying vectors by a scalar
can modify the magnitude and reverse the direction of a vector
*pg. 306 #5
don't know how to do
Scalar
has magnitude inly -EX: 7m, time
*Distributive property of addition
k(a+b)=ka+kb {k∈r}
*Cosine law
needs 2 sides and an enclosed angle
*Equal vectors
parallel with the same magnitude and direction
*Opposite vectors
parallel with the same magnitude and opposite direction
OP (R²)
position vector -O: origin (0,0) tail of vector -P: point (a,b) head of vector
OP (R³)
position vector -O: origin (0,0,0) tail of vector -P: point (a,b,c) head of vector
*Determining vector magnitude
pythagorean theorem
R²
refers to xy plane -(x,y) such that x,y∈R
R³
refers to xyz plane (3D) -(x,y,z) point on the xyz plane, x,y,z∈R -yz plane is flat on paper,x-axis is sticking out of paper
Collinear vectors
two vectors that are parallel or lie on the same straight line -can be translated so that they lie on the same straight line -vectors that are not collinear are not parallel -vectors are collinear if and only if they are scalar multiples of eachother
*Coplanar
when two or more points lie on the same plane Referring to the picture: a and b form a plane, π,if a vector c can be written as a linear combination of a and b, c is said to be coplanar
