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Mathematics · Ch 8 — Vectors

Vectors and Scalars — Magnitude and Direction

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Vectors and Scalars — Magnitude and Direction

Scalars and Vectors

Physical quantities are broadly of two kinds. A scalar quantity is completely described by a single real number together with an appropriate unit -- examples are length, mass, temperature, time and speed. A vector quantity, by contrast, needs both a magnitude and a direction to be completely specified -- examples are displacement, velocity, acceleration and force. Saying "a car travelled 6060 km" describes a scalar (distance); saying "a car travelled 6060 km due north" describes a vector (displacement).

Representing a Vector

Geometrically, a vector is represented by a directed line segment. If a vector starts at a point AA (the initial point) and ends at a point BB (the terminal point), it is written AB⃗\vec{AB} and drawn as an arrow from AA to BB. The length of the segment ABAB gives the magnitude (or modulus) of the vector, written ∣AB⃗∣|\vec{AB}| or simply ABAB, and the arrowhead gives its direction.

∣AB⃗∣=(x2−x1)2+(y2−y1)2+(z2−z1)2|\vec{AB}|=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2+(z_2-z_1)^2}

for A(x1,y1,z1)A(x_1,y_1,z_1) and B(x2,y2,z2)B(x_2,y_2,z_2).

A vector is also commonly denoted by a single arrow-topped lower-case letter, such as a⃗\vec a or b⃗\vec b (printed in bold as a\mathbf{a} in some textbooks). Its magnitude is written ∣a⃗∣|\vec a| or aa. By convention the magnitude of a vector is always a non-negative real number.

The Angle a Vector Makes with the Axes

In three-dimensional space, every nonzero vector a⃗\vec a makes some angle with each of the three coordinate axes Ox,Oy,OzOx,Oy,Oz. These three angles, each conventionally taken between 00 and π\pi (i.e. 0°0° and 180°180°), are usually denoted α,β,γ\alpha,\beta,\gamma, and are central to describing a vector's direction precisely -- they lead directly to the direction cosines taken up in the next section.

Note

Distance is always positive, but displacement (a vector) needs a direction as well -- "55 m to the right" is a different vector from "55 m to the left", even though both have the same magnitude.

Why Direction Matters

Two vectors with the same magnitude are not necessarily equal, or even comparable, unless their directions match too. A force of 1010 N pushing east and a force of 1010 N pushing west have the same magnitude but produce opposite physical effects -- which is exactly why a vector must record direction as well as size, and why vector algebra (developed through the rest of this chapter) needs its own rules of addition and multiplication, distinct from ordinary scalar algebra.

Figure 1A vector as a directed line segment

What this figure shows. A directed line segment is drawn from an initial point labelled A to a terminal point labelled B, with a single arrowhead at B showing the direction of travel from A to B; the length of segment AB represents the vector's magnitude, and the segment is labelled with an arrow-topped 'AB' symbol near its midpoint. A second panel alongside redraws the same vector starting from the origin O of a set of coordinate axes and ending at a point P, showing that a vector can be translated to any initial point without changing its magnitude or direction, and that when drawn from the origin it becomes the position vector of P.

1: A vector as a directed line segment.

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