Vector AlgebraSome Basic Concepts
Draw any straight line , in a plane or in space. Put an arrowhead on it and it becomes a directed line: it now points one way and not the other. Cut out the piece between two points A and B, keep the arrowhead, and you get a directed line segment — something with a length and a direction. That is exactly what a vector is.
Definition 1: A quantity that has magnitude as well as direction is called a vector.
The directed line segment from A to B is the vector , often written simply as . A is its initial point and B its terminal point. The distance between them is its magnitude (or length), written , or just . The arrowhead shows its direction.
Drag the initial point A and the terminal point B. The readout measures the magnitude and the direction of the vector AB. Complete all three missions.
The magnitude always came out as the square root of a sum of squares — Pythagoras, as usual — and it was never negative. A length cannot be negative, so writing makes no sense at all.
Position Vector
From Class XI you know the three-dimensional right-handed coordinate system: three perpendicular axes OX, OY and OZ through the origin O. Take any point P(x, y, z) in space. The vector that starts at the origin and ends at P is called the position vector of P with respect to O. Your third mission built one in the plane.
By the distance formula, its magnitude is
So the point P(2, 3, 6) has a position vector of magnitude
Position vectors of points A, B, C, … are usually named with the matching small letters, , , , … So "the point A with position vector " simply means .
Direction Cosines
The position vector of a point P(x, y, z) points somewhere in space. To say where, measure the angles it makes with the positive directions of the axes: with the x-axis, with the y-axis and with the z-axis. These are its direction angles, and their cosines
are its direction cosines.
This is Fig. 10.3, and you can turn it: drag it sideways. Pick a point and an axis. The dashed drop from P meets the axis at A at a right angle, so the amber piece OA is the shadow of OP on that axis.
In the right-angled triangle OAP, , that is, . So
and the coordinates of P can be written as . For P(2, 3, 6), where r = 7, the direction cosines are ,
The numbers , , , which are proportional to the direction cosines, are called direction ratios of the vector, usually written a, b and c. For P(2, 3, 6) the direction ratios 2, 3, 6 are simply its coordinates.
Note
Squaring and adding the direction cosines always gives 1, because :
Direction ratios obey no such rule: for 2, 3, 6 the sum of the squares is