Vector Basics Flashcards
Vector definition
A direction in space with a magnitude
What does it mean for a vector to be bound?
A vector with a specific starting or finish or point on it.
Representation of vectors
V = u1i + u2j + u3k
or drawn on top of one another
Magnitude of a vector
Pythagoras of coefficients:
|v|=(v1^2 + v2^2 +v3^2)^1/2
Unit vector
Unit vector doesn’t have a magnitude so is just a direction:
Unit v = v/|v|
Adding vectors
a±b =( a1±b1)i +( a2±b2)j +( a3±b3)k
Multiplication of vectors by a scalar
We multiply each coefficient by the scalar
Creating a vector between two points
Take away one point from the other each coord e.g.
v = (x1-x0)i + (y1-y0)j +(z1-z0)k
Cross product
a x b = Do matrix determinant equation with +I -J +K and cancel lines
Dot product
a.b = (a1b1)j+ (a2b2)k+(a3b3)k
Angle between 2 vectors with cos(theta)
a.b = |a||b| cos(theta), where theta is the angle between a and b
Orthogonal definition
Vector is perpendicular to the other vector
Sin equation with cross product
a x b = |a||b| ñ sin(theta)
Theta is the angle between a and b and n is the unit vector of a x b
uA x hB =
uh(A xB)
A x (B+C) =
A x B + A x C
Equation of a line
r = ro +hv
where r is a generic point, ro is a given point and v is the vector for that equation and h is scalar multiplying v.
We can split into each axis e.g. x, y and z
What do we need for equation of a line
2 points or a point and a vector
Equation of a plane
r =ro +hu +gv
Where r is a point on the plane, ro is a given point, h and g are scalrs and u and v are vectors of the plane.
or Ax+By+Cz =d
What do we need for equation of a plane
point and a vector, 3 points or 2 points and a vector
If a point lies on a plane, vector between it and ro will be…
perpendicular
