Vectors

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Vectors is the second lecture in the Mechanics section of PH1011. It covers an introduction to what vectors are and how they can be combined through addition/subtraction and multiplication.

Next: Motion in 2D and 3D

Vectors
Vectors are those quantities which are described by both a magnitude and direction; they are represented visually by an arrow and can act in two or three dimensions. The direction is defined as angle θ counter clockwise from the x axis, and the magnitude is given as |a| = √(a.a) = √(ax2 + ay2 + az2).

Any vector can be resolved into components acting horizontally and vertically; this allows calculation of its angle and/or magnitude through trigonometry. The components of a vector are scalar quantities.

Unit vectors are vectors with a magnitude of 1 acting along axis: i acts on the x axis, j acts on the y axis and k acts on the z axis. Resultant vectors can be expressed as a multiple of a combination of unit vectors.

Mathematical Combination
Addition and subtraction of vectors is most easily achieved graphically; placing the vectors nose to tail for addition and tail to tail for subtraction allows the resultant vector to be found. Multiplying a vector by a scaler creates a resultant with the same direction but multiplied magnitude to its original.

Multiplication of two vectors can result in a scalar; a.b = |a|.|b|cosθ = axbx + ayby + azbz. The scaler product of perpendicular vectors is 0.

Summary
Vectors have magnitude and direction, and can be added graphically or multiplied to give scalars. Unit vectors act on axis and have a magnitude of one.