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