Please enable JavaScript.
Coggle requires JavaScript to display documents.
Vector (Notation (Vector (< a, b, c >, ai + bj + ck), Absolute sign…
Vector
Notation
Point
P(x , y, z)
Vector
< a, b, c >
a
i
+ b
j
+ c
k
Absolute sign
Constant
|
u
|
Norm / Magnitude
||
u
||
Cross product
Find new vector that is orthogonal to
A
snd
B
Formula
Determinant
A
x
B
= ( ||
A
|| ||
B
|| sinθ)
n
||
A
x
B
||
Find the area
triangle
1/2 ||
A
x
B
||
parallelogram
||
A
x
B
||
Find a volume of parallelepiped
|
u
. (
v
x
w
) |
Coplanar
|
u
. (
v
x
w
) | = 0
Line & Plane
Line
Conditions
At least one point on line
Vector
parallel
to line
Line equation
Parametric equation
x = x0 + at
y = y0 + at
z = z0 + at
Symmetric equation
x - x0 / a = y - y0 / b = z - z0 / c
Plane
Conditions
At least on point on plane
Vector
perpendicular
to plane
Plane equation
a(x - x0) + b(y - y0) + c(z - z0) = 0
ax + by + cz + d = 0
d = -ax0 - by0 - cz0
Line intersection between 2 planes
Method 1
cut off one variable
let one variable equal to t
substitute to other two variables
Method 2
line intersection //
n1
x
n2
Example
2x - y + z = 1
n1
= < 2, -1, 1 >
x + y - 3z = 2
n2
= < 1, 1, -3 >
n1
x
n2
= < 2, 3, 7>
let one variable to 0
Find a vector
Locating the point
Vector between 2 points
Operator
Plus (+) / Minus (-)
Multiple
by real number
Parallel vector
u = kv
k = u / v
Component & Projection
Component (Scalar) of
A
on
B
||
A
|| cosθ
A
.
B
/ ||
B
||
Projection (Vector) of
A
on
B
(Comp
A
on
B
)(unit vector of
B
)
Find magnitude of vector
Unit vector
magnitude = 1
unit vector = u / ||u||
Dot product
A . B
||
A
||||
B
||cosθ
a1b1 + a2b2 + a3b3
Classify the angle
A . B > 0
acute
A . B = 0
orthogonal
A . B < 0
obtuse
Work done by force
W = FS
F
.
d
= ||
F
||||
d
||cosθ