More about vectors
Coplanar vectors
Q
A
B
Ca
b
c QA = λa
QB = μb
c = QA + QB
c = λa + μb a, b, and c are coplanar vectors
a = (1, 1)
b = (2, 1)
c = (3, 4)
λ = ?
μ = ?
5
-1
Coplanar vector in 3D
z
x
21
O
1 2 31
2
3
34
ac
b
c = λa + μb
Position vector A vector which has its initial point at the origin of
coordinates.
x
y
yQ
xQ
r
rx
ry
Q (xQ, yQ)
O
z
x
21
O
1 2 31
2
3
34
rP
2
3
P (2, 3, 3)
PrOP = (2, 3, 3)
r = (xQ, yQ)
Vector equations for curves & surfaces Circle
22 )()( byax
y
rC
rC (a, b)
O
P (x, y)
r - rC
r – rC = c
Proof: r – rC = (x, y) – (a, b) = (x-a, y-b)
r - rC
222 )()( cbyax
Vector equations for curves & surfaces Plane
z
x
O
A
C
B
AP = λAB + μAC x
rA rB
rC
rP
AB = rB - rA
AC = rC - rA
AP = r - rA
r – rA = λ (rB - rA)+ μ(rC – rA)
-- Parametric vector equation for a plane
Example-plane A (1,2,1), B(2,2,0), C(2,1,2)
(x, y, z) –(1, 2, 1) = λ{(2, 2, 0)-(1, 2, 1)} + μ{(2, 1, 2)-(1, 2, 1)}
r – rA = λ (rB - rA)+ μ(rC – rA)
= λ(1, 0, -1) + μ(1, -1, 1)
x = 1 + λ+ μ
y = 2 - μ
z = 1 – λ + μ x + 2 y + z = 6
Cartesian parametric equations for the plane
Cartesian (general) equation for a plane
(x, y, z) = (1+λ+μ, 2-μ, 1-λ+μ)
Vector parametric equation for a plane
1c’ d’
b’ d’
Plane cont’d
General form of a plane
A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3)
a
a
a
a’ d’x y z = + + cba
3
2
1
x
x
x
3
2
1
y
y
y
3
2
1
z
z
z
b
b
b
c
c
c
1
1
1
c
b
a
0d
General Cartesian form of a plane
A(x1, y1, z1), B(x2, y2, z2), C(x3, y3, z3)
Plane cont’d
a
a
a
dx y z = + + cba
3
2
1
x
x
x
3
2
1
y
y
y
3
2
1
z
z
z
b
b
b
c
c
c
d
d
d
dC
dC
dC
c
b
a
3
2
1
General form of a plane
Special cases:
d=0: ax+by+cz=0a plane passing through the origin
a=0: by+cz=da plane parallel to the x axis
b=0: ax+cz=da plane parallel to the y axis
c=0: ax+by=da plane parallel to the z axis
Special planes
dx y z = + + cba
Vector equations for curves & surfaces Line
z
x
O
C
A
AP = λACx
rA
rC
rP
AC = rC - rA
AP = r - rA
r – rA = λ (rC – rA)
-- Parametric vector equation for a line
Example—Line
A(1,2,1), C(3, 0, -1)
2
1
2
2
2
1 zyx
z
x
O
C
A
x
rA
rC
rP
r – rA = λ (rC – rA)
(x, y, z)-(1, 2, 1)=λ{(3, 0, -1)-(1, 2,1)}
(x, y, z)=(1+2λ,2-2λ, 1-2λ)
21
22
21
z
y
x
AP = λ AC
Unit vector A vector of unit magnitude
e.g.
7
6,
7
3,
7
2a
17
6
7
3
7
2222
a
What about the vectors, b=(1,0,0), c=(0,1,0), d=(0,0,1)?
If a is a vector, then… the unit vector in the direction of a =(ax, ay, az) is:
a
aa ˆ
aa
z
x
y
θz
θx
θy
aaazyx aaa
,,
zyx cos,cos,cos
Basis vectors
i = (1, 0, 0) j = (0, 1, 0) k = (0, 0, 1) Any vector, a = (a1, a2, a3)
can be written as: a = a1 +a2 +a3
ki
j
x
y
z
1
1
1i j k
a = (a1, a2, a3) = (a1, 0, 0)+(0, a2, 0)+(0, 0, a3)
= a1(1, 0, 0)+a2 (0, 1, 0)+a3(0, 0, 1)
i j k = a1 +a2 +a3