Theoretical Mechanics/Vector Algebra

Scalars and vectors

Scalars

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

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.

Basis Vectors

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 e_x, e_y, e_z.

You may also see $\hat{\imath}$ for e_x, $\hat{j}$ for e_y, and $\hat{k}$ for e_z.

Every vector is then a linear combination of these basis vectors. Often, vectors are written as tuples, so that the vector ae_x + be_y + ce_z 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 = 3e_x + 4e_y
length: |V| = sqrt (3² + 4²) = sqrt (9 + 16) = sqrt 25 = 5
Unity vector e_v = (3/5)e_x + (4/5)e_y

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.

Equality

Two vectors are equal if and only if both their magnitude and their direction are the same.
4e_x ≠ 5e_x
4e_x ≠ 4e_y
4e_x + 8e_y - 1e_z ≠ 4e_x + 8e_y + 3e_z
4e_x = 4e_x
10e_y = 10e_y
4e_x + 8e_y - 1e_z = 4e_x + 8e_y - 1e_z

Vector multiplied with scalar

Multiply the magnitude of the vector with the scalar.
4e_x + 3e_y* 2 = 8e_x + 6e_y
3e_a * 5 = 15e_a

Angle between 2 vectors

This is the smallest angle between both vectors.

Adding Two Vectors

Two vectors are added component by component, like this:

(a₁e_x + a₂e_y + a₃e_z) + (b₁e_x + b₂e_y + b₃e_z) = (a₁ + b₁)e_x + (a₂ + b₂)e_y + (a₃ + b₃)e_z

So, for example:

(2e_x + 1e_y + 3e_z) + (e_x + 1e_y + e_z) = (3e_x + 2e_y + 4e_z)

Subtracting Two Vectors

Subtracting a vector A from a vector B is no different than adding A and -1 * B together.

Dot product

a*b*cos θ
With: a, b magnitude of the vectors; θ angle between the vectors.

A.B = a_x * b_x + a_y * b_y + a_z * b_z

with A,B 2 vectors; a_x 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°

Cross product

U = A x B
U = a * b * sin θ
U = (A_y * b_z - a_z * b_y) e_x
+ (A_z * b_x - a_x * b_z) e_y
+ (A_x * b_y - a_y * b_x) e_z

Properties of the cross product: Is a vector orthogonal to both A and B