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, 5 kg, 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 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. Equality 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. Dot product 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 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