Lecture 1: 13.1 13.2 Flashcards
A two dimensional vector is notated as?
V = (P Q)->, where P = initial point and Q = terminal point.
V is a vector with an arrow pointing from?
P to Q (initial point to terminal)
Initial point and terminal point has coordinates?
(X, Y)
What is a position vector?
A vector from the origin to a point, R.
Parallel vectors?
Vectors of nonzero length if the lines through v and w are parallel. They do not need to point in the same direction.
Translation in vectors?
When a vector is moved without changing the magnitude and direction (different base point).
Equivalent vectors are those that?
Have the same direction and magnitude.
Every vector v is or is not equivalent to a position vector?
Is
Components of a Vector (a1, b1) and Q = (a2, b2), are the quantities
a = a2 − a1 (x-component), b = b2 − b1 (y-component)
The pair of components is denoted by?
⟨a, b⟩.
Zero vector?
A vector with no length and without direction.
Vector sum v + w is defined when?
v and w have the same base point.
What is x + w when the tail of w is on the head of x?
It is the vector from the tail of x and the head of w.
Parallelogram law?
v + w if the tail of x and w are the same but pointing different directions, the v + w vector will have the same tail but will point exactly in the middle of v and w.
How to add several vectors?
Translate each position vector in which the terminal is the initial of the next vector. v = v2 + v2 + … + v_n. The new vector, v will be a position vector pointing to the terminal of the last vector, v_n.
Vector subtraction?
v − w is carried out by adding −w to v.
∥λv∥ = ?
|λ| ∥v∥
A vector w
is parallel to v if and only if?
w = λv for some nonzero scalar λ.
Vector v points in the same/opposite direction if lambda > 0 and same/opposite direction if lambda < 0?
Same, different.
Vector Operations Using Components: If v = ⟨a, b⟩ and w = ⟨c, d⟩, then for addition, subtraction, coefficients, and addition of 0? 4•
(i) v+w=⟨a+c,b+d⟩
(ii) v−w=⟨a−c,b−d⟩
(iii) λv = ⟨λa, λb⟩
(iv) v+0=0+v=v
if P = (a1, b1) and Q = (a2, b2)?
PQ->=OQ->−OP-> =⟨a2,b2⟩−⟨a1,b1⟩=⟨a2 −a1,b2 −b1⟩
THEOREM 1 Basic Properties of Vector Algebra For all vectors u, v, w and for all
scalars λ? Commutative law, associative law, distributive law? 3•
Commutative Law: v + w = w + v
Associative Law: u + (v + w) = (u + v) + w
Distributive Law for Scalars: λ(v + w) = λv + λw
A linear combination of vectors v and w is a vector?
rv + sw
Linear combination as system of linear equations?
{rv1 + sw1 = u1
{rv2 + sw2 = u2
e_v is?
A unit vector in the direction v
If i = ⟨1,0⟩, j = ⟨0,1⟩, then v, which is the linear combination vector, is?
v = ⟨a, b⟩ = ai + bj, in which a and b are scalers of i and j.
Octant?
Is one of the eight divisions in a 3d coordinate space divided by four planes throughout each one of the axis.
Vector addition is commutative, is associative, and satisfies the distributive property
with respect to scalar multiplication?
Is commutative, is associative, and satisfies the distribution property.
