Many measurements in biology and in the laboratory are fully described by one number: the mass of a leaf sample, the volume of a buffer or the temperature of a water bath. Other quantities also have a direction, and when the movement or the push happens on a flat surface such as a field plot, a microscope slide or a bench top, the direction needs a second number. This topic introduces the flat (two-dimensional) setting in which vectors and scalars are used in the rest of the chapter.
A two-dimensional space is a flat surface in which any position can be fixed by two numbers. Two perpendicular number lines, called axes, cross at a point called the origin. The horizontal axis is the x-axis and the vertical axis is the y-axis. On a field plan the x-axis is usually taken to point east and the y-axis north; on a slide or a graph sheet, x points to the right and y points up.
A point is written as an ordered pair (x, y). The first number is the distance along the x-axis and the second is the distance along the y-axis, each measured from the origin. Positive values lie to the right of, or above, the origin and negative values lie to the left of, or below, it.
Example: a plot peg is placed 4 m east and 3 m north of the corner peg that is used as the origin. Its position is (4, 3). A second peg placed 2 m west of the corner and 5 m north is at (-2, 5).
A scalar quantity is stated by one number and a unit and has no direction, so the plane makes no difference to it: the mass of a seed sample is 12.4 g whether the sample sits at one end of the bench or the other, and 250 mL of nutrient solution is 250 mL wherever the bottle stands. Scalars are defined, and listed with further examples, in Definitions of vector and scalar quantities.
A vector quantity needs a size and a direction. In a plane the direction is carried by two components: how far (or how fast, or how hard) the quantity acts along the x-axis and how far along the y-axis. A vector is therefore written as a pair of numbers in brackets, in the same order as a point: (x, y). Written as a column, the same vector is
a = | 3 |
| 4 |
The top number is the x-component and the bottom number is the y-component. The unit vectors i = (1, 0) and j = (0, 1) point one unit along the x-axis and the y-axis, so the vector above can also be written a = 3i + 4j.
Example: seeds released from a plant are carried by the wind at 3 m/s towards the east and 2 m/s towards the north. One number cannot describe this drift, because a speed of "3 m/s" does not say where the seeds land. The wind velocity is the vector (3, 2) m/s, and it tells both how fast and which way.
On a straight line, direction is a matter of sign: a stream-level logger that moves +12 cm (up) or -5 cm (down) needs only a plus or a minus. In a plane, a plus or a minus is not enough. A force of 6 N pulling a clip towards the right and slightly upwards cannot be described by a sign, so it is described by its two components, for example (5.5, 2.4) N.
Two vectors are equal when both their components are equal. The vectors (3, 4) and (4, 3) are different, even though they use the same numbers, because the order carries the direction.
Example: a floating mat of algae on a pond drifts 6 m west and 2 m north in one hour. Taking east and north as positive, the x-component is -6 m and the y-component is +2 m, so the drift is (-6, 2) m.
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