Summary for §9.1 - §9.5
Math 1321 (Qinghai Zhang)
Definition 1. The Cartesian product X × Y between
two sets X and Y is the set of all possible ordered pairs
with first element from X and second element from Y :
X × Y = {(x, y) | x ∈ X, y ∈ Y }.
(1)
2013-FEB-01
Theorem 12. If u, v, w ∈ R3 , and c ∈ R, then
u · u = |u|2
(9a)
u·v =v·u
(9b)
u · (v + w) = u · v + u · w
(9c)
Definition 2. A 3D coordinate system is a one-to-one
(cu) · v = c(u · v) = u · (cv)
(9d)
correspondence between spatial points P and ordered
0·u=0
(9e)
triplets (x, y, z) ∈ R3 . The numbers x, y, z are called
the coordinates.
Definition 13 (Projection). The vector projection of b
onto
a is
Formula 3. The distance between two points A =
a
proja b = ca = (compa b)
,
(10)
(xA , yA , zA ) and B = (xB , yB , zB ) is
|a|
p
|AB| = (xA − xB )2 + (yA − yB )2 + (zA − zB )2 . (2) where the scalar projection of b onto a is
Definition 4. A vector v = hvx , vy , vz i is a geometric
a·b
compa b =
.
(11)
object that has both a magnitude and a direction. Its
|a|
magnitude is |v| = |OPv |, where Pv = (vx , vy , vz ) and O
is the origin of the 3D coordinate system. Its direction Definition 14. A plane is a set of points uniquely deis determined by the arrow from O to Pv . The numbers termined by a point P0 and a normal vector n:
vx , vy , vz are called the components.
{P | n · (P − P0 ) = 0}.
(12)
Axiom 5 (Vector addition and scaling). If c ∈ R,
Equivalently, the scalar equation of a plane is
u = hxu , yu , zu i, v = hxv , yv , zv i, then
ax + by + cz + d = 0.
u + v = hx + x , y + y , z + z i ,
(3a)
u
v
u
v
u
v
(13)
(3b) Definition 15 (cross product: algebraic definition).
The cross product of a, b ∈ R3 is a vector
i
j
k
(4a)
a
a
a
a × b = det 1
(14a)
2
3 (4b)
b1 b2 b3 (4c)
a2 a3 a1 a3 a1 a2 (4d) = det b2 b3 i − det b1 b3 j + det b1 b2 k
cv = hcvx , cvy , cvz i .
Axiom 6. If c ∈ R and u, v, w ∈ Rn , then
u + v = v + u,
u + (v + w) = (v + u) + w,
u+0=u
u + (−u) = 0
c(u + v) = cv + cu,
(4e)
(14b)
(c + d)u = cu + du
(4f)
(cd)u = c(du),
(4g)
= (a2 b3 − a3 b2 )i − (a1 b3 − a3 b1 )j + (a1 b2 − a2 b1 )k
(14c)
1u = u.
(4h) Theorem 16. If c ∈ R and u, v, w ∈ R3 , then
Definition 7. The standard basis vectors in R are
u × v = −v × u
i = h1, 0, 0i ,
j = h0, 1, 0i ,
k = h0, 0, 1i . (5)
(cu) × v = c(u × v) = u × (cv)
(15b)
Definition 8. v is a unit vector iff |v| = 1. The unit
v
.
vector in the same direction of a vector v is |v|
u × (v + w) = u × v + u × w
(15c)
(u + v) × w = u × w + v × w
(15d)
3
Definition 9. A line is a set of points uniquely determined by a point P0 = (x0 , y0 , z0 ) and a direction vector
v = ha, b, ci:
{P | P (t) = P0 + tv, t ∈ (−∞, +∞)}.
(15a)
Definition 17. Two nonzero vectors u, v are parallel if
and only if ∃c 6= 0, s.t. u = cv. Alternatively, u, v are
parallel if and only if u × v = 0.
(6)
Definition 18 (scalar triple product). For a, b, c ∈ R3 ,
Definition 10 (Dot product: algebraic definition). The
a1 a2 a3 dot product of two vectors a, b ∈ R3 is a real number:
a · (b × c) = (a × b) · c = det b1 b2 b3 . (16)
c1 c2 c3 a · b = a 1 b1 + a 2 b2 + a 3 b3 .
(7)
Definition 11. The angle θ between two nonzero vec- Definition 19 (vector triple product). For a, b, c ∈ R3 ,
tors a, b ∈ R3 satisfies
a × (b × c) = (a · c)b − (a · b)c.
(17)
a·b
cos θ =
,
θ ∈ [0, π].
(8)
|a||b|
1