Vector
Notation
Point
Vector
Absolute sign
Constant
Norm / Magnitude
| u |
|| u ||
Find a vector
Locating the point
Vector between 2 points
Operator
Plus (+) / Minus (-)
Multiple
by real number
Parallel vector
u = kv
k = u / v
Find magnitude of vector
Unit vector
magnitude = 1
unit vector = u / ||u||
Component & Projection
Component (Scalar) of A on B
Projection (Vector) of A on B
Dot product
Cross product
Line & Plane
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
parallelogram
|| A x B ||
1/2 || A x B ||
Find a volume of parallelepiped
| u . ( v x w ) |
Coplanar
| u . ( v x w ) | = 0
|| A || cosθ
A . B / ||B||
(Comp A on B)(unit vector of B)
Work done by force
W = FS
F . d = ||F||||d||cosθ
Line
Conditions
At least one point on line
Vector parallel to line
Line equation
Parametric equation
Symmetric equation
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
x = x0 + at
y = y0 + at
z = z0 + at
x - x0 / a = y - y0 / b = z - z0 / c
P(x , y, z)
< a, b, c >
ai + bj + ck
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
A . B
||A||||B||cosθ
a1b1 + a2b2 + a3b3
Classify the angle
A . B > 0
A . B = 0
A . B < 0
acute
orthogonal
obtuse