Theoretical Mechanics/Vector Algebra

From Wikibooks, open books for an open world
< Theoretical Mechanics
Jump to: navigation, search

Scalars and vectors[edit]


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.

Basis Vectors[edit]

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 \hat{\imath} for ex, \hat{j} for ey, and \hat{k} 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[edit]

Multiply the magnitude of the vector with the scalar.
4ex + 3ey * 2 = 8ex + 6ey
3ea * 5 = 15ea

Angle between 2 vectors[edit]

This is the smallest angle between both vectors.

Adding Two Vectors[edit]

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[edit]

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

Dot product[edit]

a*b*cos θ
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°

Cross product[edit]

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