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The concept of dot products and cross products between vectors, their properties, and their applications in finding distances to planes, normal vectors of triangles, and area of triangles. It also includes a vector3 class implementation for vector operations.
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The dot product is a scalar value that tells
us something about the relationship
between two vectors
If a b > 0 then _ < 90º
If a b < 0 then _ > 90º
If a b = 0 then _ = 90º (or one or more of
the vectors is degenerate (0,0,0))
a
u
a u
If | u |=1.0 then a · u is the length of the projection
of a onto u
The distance is the length of the projection
of x - p onto n :
p
n
- x
x-p
dist = ( x − p )⋅ n
Dot Products with Unit Vectors
b
_ a
a b = 0
0 < a b <
a b = -
a b = 1
-1 < a b <
cos ( θ )
a
b
0
sin
× =
× =
× =
×
a b
a b
a b a b
a b
θ
area of parallelogram ab
is a vector perpendicular to both
a and b , in the direction defined by
the right hand rule
if a and b are parallel
Find the unit length normal of the triangle
defined by 3D points a , b , and c
a
b
c
Find the area of the triangle defined by 3D
points a , b , and c
a
b
c
= ( b − a )× ( c − a )
b-a
c-a
a
b
c
Example: Alignment to Target
p
h
t t-p
_
a
( )
( )
( )
( )
−
1
θ cos
Vector Class
class Vector3 {
public:
Vector3() {x=0.0f; y=0.0f; z=0.0f;}
Vector3(float x0,float y0,float z0) {x=x0; y=y0; z=z0;}
void Set(float x0,float y0,float z0) {x=x0; y=y0; z=z0;}
void Add(Vector3 &a) {x+=a.x; y+=a.y; z+=a.z;}
void Add(Vector3 &a,Vector3 &b) {x=a.x+b.x; y=a.y+b.y; z=a.z+b.z;}
void Subtract(Vector3 &a) {x-=a.x; y-=a.y; z-=a.z;}
void Subtract(Vector3 &a,Vector3 &b) {x=a.x-b.x; y=a.y-b.y; z=a.z-b.z;}
void Negate() {x=-x; y=-y; z=-z;}
void Negate(Vector3 &a) {x=-a.x; y=-a.y; z=-a.z;}
void Scale(float s) {x=s; y=s; z*=s;}
void Scale(float s,Vector3 &a) {x=sa.x; y=sa.y; z=s*a.z;}
float Dot(Vector3 &a) {return xa.x+ya.y+z*a.z;}
void Cross(Vector3 &a,Vector3 &b)
{x=a.yb.z-a.zb.y; y=a.zb.x-a.xb.z; z=a.xb.y-a.yb.x;}
float Magnitude() {return sqrtf(xx+yy+z*z);}
void Normalize() {Scale(1.0f/Magnitude());}
float x,y,z;
};
