Theoretical Mechanics/Vector Algebra
Scalars and vectors
Scalars are simple values. They describe something like height, length, mass, number of chairs with a real number. e.g. 10 chairs, 5Kg, 2m: 10, 5 and 2 are scalars.
Vectors have a magnitude and a direction. e.g. the velocity of a ball has both a speed and a direction in which it is moving.
All cartesian coordinate systems have a number of basis vectors equal to their dimension; for most applications in Theoretical Mechanics the vector space is E3, whose basis vectors are ex, ey, ez.
You may also see for ex, for ey, and for ez.
Every vector is then a linear combination of these basis vectors. Often, vectors are written as tuples, so that the vector aex + bey + cez is equivalent to the tuple (a, b, c).
The unity vector of any vector can be found by dividing all of its components by the length:
V = 3ex + 4ey
length: |V| = sqrt (3² + 4²) = sqrt (9 + 16) = sqrt 25 = 5
Unity vector ev = (3/5)ex + (4/5)ey
3/5 equals to the cosinus of the angle of this vector with the X-axis, 4/5 is the cosinus of the angle of this vector with the Y-axis:
argcos(3/5) = about 53°
argcos(4/5) = about 37°
Notice that both angles add up to 90°. This means this vector lays in the XY-plane. If it adds up to a larger number, it lays inside a 3D volume, not a plane.
Two vectors are equal if and only if both their magnitude and their direction are the same.
4ex ≠ 5ex
4ex ≠ 4ey
4ex + 8ey - 1ez ≠ 4ex + 8ey + 3ez
4ex = 4ex
10ey = 10ey
4ex + 8ey - 1ez = 4ex + 8ey - 1ez
Vector multiplied with scalar
Multiply the magnitude of the vector with the scalar.
4ex + 3ey * 2 = 8ex + 6ey
3ea * 5 = 15ea
Angle between 2 vectors
This is the smallest angle between both vectors.
Adding Two Vectors
Two vectors are added component by component, like this:
(a1ex + a2ey + a3ez) + (b1ex + b2ey + b3ez) = (a1 + b1)ex + (a2 + b2)ey + (a3 + b3)ez
So, for example:
(2ex + 1ey + 3ez) + (ex + 1ey + ez) = (3ex + 2ey + 4ez)
Subtracting Two Vectors
Subtracting a vector A from a vector B is no different than adding A and -1 * B together.
With: a, b magnitude of the vectors; θ angle between the vectors.
A.B = ax * bx + ay * by + az * bz
with A,B 2 vectors; ax X-component of vector A, ...
Properties of the dot product: Is a scalar. When equal to zero: The angle between the vectors is 90°. or one of the vectors is zero. Maximum: angle between the vectors is 0°
U = A x B
U = a * b * sin θ
U = (Ay * bz - az * by) ex
+ (Az * bx - ax * bz) ey
+ (Ax * by - ay * bx) ez
Properties of the cross product: Is a vector orthogonal to both A and B