Numbers With Somewhere to Go Vector Algebra

Ask someone how fast a car is going and a single number answers it completely: 60 km/h. That is a scalar, a quantity fully described by magnitude alone, with nothing left unsaid. Now ask which way a hiker walked: '3 km' does not finish the sentence, because 3 km north and 3 km south leave the hiker in completely different places. A quantity that needs both a magnitude AND a direction to be fully described is called a vector. Geometrically, a vector is drawn as a directed line segment: an arrow starting at an initial point and ending at a terminal point, with the arrowhead showing direction and the segment's length showing magnitude. A vector from point A to point B is written AB (with the arrow understood) or often just as a single lowercase letter such as a. Its magnitude, the plain distance between A and B, is written |AB| or |a|, and because a length can never be negative, an expression like |a| < 0 simply can never happen, by definition. Since a vector can be slid anywhere in space without changing its magnitude or direction, the vectors used throughout this chapter are understood to be free vectors: only length and direction matter, never the specific spot where the arrow happens to be drawn.

Once three perpendicular axes (x, y, z) are set up with a fixed origin O, every point P in space can be pinned down by a single vector: the position vector OP, running from the origin straight to P. If P has coordinates (x, y, z), the magnitude of its position vector follows directly from the distance formula: |OP| = sqrt(x^2 + y^2 + z^2). Position vectors of several points are usually written using the point's own letter, a for the position vector of A, b for B, and so on, which turns geometry problems about points into algebra problems about vectors. Beyond just locating a point, a position vector also has an orientation, described by the angles it makes with each axis. The angles alpha, beta, gamma that OP makes with the positive x, y and z axes respectively are its direction angles, and their cosines, cos(alpha), cos(beta), cos(gamma), usually named l, m, n, are its direction cosines. These three numbers always satisfy l^2 + m^2 + n^2 = 1, no matter what the vector is, because they come from a single length-1 relationship built into the geometry. Any three numbers merely proportional to l, m, n (found by scaling by the vector's own length r) are called direction ratios, usually named a, b, c, and unlike direction cosines, direction ratios do NOT need to satisfy a^2 + b^2 + c^2 = 1 in general, since they are just one of infinitely many proportional scalings.

A handful of named vector types come up constantly and are worth recognising on sight. A zero (or null) vector has its initial and terminal points coinciding, so its magnitude is exactly 0; because a vector of length 0 has nowhere to point, its direction is left undefined, or, equivalently, treated as arbitrary. A unit vector has magnitude exactly 1; the unit vector pointing the same way as a given nonzero vector a is written with a small cap and obtained by dividing a by its own magnitude. Vectors sharing the same initial point are called coinitial vectors. Vectors that are all parallel to one common line, regardless of their individual magnitudes or which way along that line they point, are called collinear vectors. Two vectors are equal only when they share both the same magnitude AND the same direction; their initial points do not need to match, since free vectors can be slid around freely. Finally, the negative of a vector a is a vector with the identical magnitude but the exactly opposite direction, written -a; for instance if a runs from A to B, then -a is the vector running from B back to A. Notice the layered relationship here: all equal vectors are automatically collinear, since they share a direction, but collinear vectors need not be equal, since they might differ in magnitude or point opposite ways along that same line.

Suppose a delivery drone flies from point A to point B, then from B onward to C. Its net displacement is simply the vector from A to C, and this is exactly the triangle law of vector addition: AB + BC = AC. To add any two vectors a and b in general, slide b (keeping its magnitude and direction fixed) so its initial point sits on the terminal point of a; the vector from a's initial point to b's new terminal point is the sum a + b. A useful special case follows at once: since AB + BC + CA returns to the starting point, this three-term sum equals the zero vector whenever the vectors form a closed triangle. An equivalent way to add two vectors, useful when they share a starting point, like two forces pulling on one object, is the parallelogram law: draw a and b as adjacent sides of a parallelogram from their common point, and their sum is the diagonal through that point. Vector addition obeys two familiar properties: it is commutative (a + b = b + a, both sums trace the same diagonal) and associative (a + (b + c) = (a + b) + c, so brackets can be dropped and the sum written as a + b + c). The zero vector is the additive identity (a + 0 = a), and subtraction a - b means adding the negative, a + (-b).

Multiplying a vector a by an ordinary number (scalar) lambda produces another vector, lambda a, that stays collinear to a; it gets stretched, shrunk, or flipped. Its magnitude becomes |lambda| times the original, |lambda a| = |lambda| |a|, and its direction stays the same as a when lambda is positive but flips when lambda is negative. Taking lambda = -1 recovers the negative vector -a, and taking lambda = 1/|a| produces a vector of magnitude exactly 1, the unit vector in the direction of a. This scaling idea combines with three special unit vectors i, j, k, pointing along the positive x, y and z axes, each of magnitude 1. Any position vector r of a point P(x, y, z) can now be written in component form as r = x i + y j + z k, where x, y, z (the scalar components) are simultaneously the point's coordinates AND the projections of r along the three axes. Applying Pythagoras' theorem twice, once in the xy-plane and once lifting to z, gives the magnitude directly from the components: |r| = sqrt(x^2 + y^2 + z^2). With vectors written this way, algebra takes over from geometry: component-form vectors are added or subtracted by combining matching components, scaled by multiplying every component by lambda, and are equal precisely when all three components agree. A tidy bonus follows: two vectors are collinear exactly when one's components are a constant multiple of the other's, matching ratio for ratio.

Given any two points P1(x1,y1,z1) and P2(x2,y2,z2), the vector running from P1 to P2 can be built by routing through the origin: P1P2 = OP2 - OP1, which works out in components to (x2-x1) i + (y2-y1) j + (z2-z1) k, simply subtracting corresponding coordinates, exactly matching the familiar distance-formula pattern, since |P1P2| = sqrt((x2-x1)^2+(y2-y1)^2+(z2-z1)^2). A closely related question is more subtle: given points P and Q with position vectors a and b, where does a third point R sit if it splits the segment PQ in some chosen ratio m:n? When R divides PQ internally, sitting between P and Q, its position vector works out to r = (m b + n a) / (m+n); when R divides PQ externally, sitting beyond one of the two points on the extended line, the position vector instead becomes r = (m b - n a) / (m-n). Setting m = n in the internal case immediately gives the single most-used special case: the midpoint of PQ has position vector r = (a+b)/2, the plain average of the two endpoints' position vectors. This section formula converts what would otherwise be a construction-and-measurement problem into a one-line vector calculation, and it reappears constantly whenever a problem asks about a point dividing a segment in a given ratio.

Multiplying two vectors together can be done in two different ways, and the first is the scalar (or dot) product, written a.b, which always produces a plain number, never a vector. For two nonzero vectors a and b separated by angle theta, the dot product is defined as a.b = |a| |b| cos(theta). This single definition packs in several facts at once. When theta = 0, cos(theta) = 1, so a.a = |a|^2. When a and b are perpendicular, cos(90 degrees) = 0, forcing a.b = 0, and conversely, whenever a.b = 0 for two nonzero vectors, they must be perpendicular, making the dot product the standard test for perpendicularity. The dot product is commutative (a.b = b.a) and distributes over addition. Writing vectors in component form turns the geometric definition into pure arithmetic: for a = a1 i+a2 j+a3 k and b = b1 i+b2 j+b3 k, the dot product simplifies to a.b = a1 b1 + a2 b2 + a3 b3, using only the fact that i, j, k are mutually perpendicular unit vectors. This also unlocks the angle itself: cos(theta) = (a.b)/(|a||b|). A closely related idea is projection: the projection of a onto a directed line (or onto another vector b) measures how much of a's length lies along that direction, computed as a dotted with the unit vector along the line, and it can be positive, negative, or zero depending on whether a leans toward, away from, or perpendicular to the line.

The second way to multiply two vectors is the vector (or cross) product, a x b, which unlike the dot product produces another VECTOR, perpendicular to both a and b. For two nonzero, non-parallel vectors at angle theta, a x b is defined as |a| |b| sin(theta) times a unit vector n perpendicular to both, its direction fixed by the right-hand rule: curl the right hand's fingers from a toward b, and the thumb points along n. Swapping the order reverses the curl, so b x a = -(a x b); the cross product is NOT commutative. Whenever a and b are parallel or collinear, theta is 0 or 180 degrees, so sin(theta) = 0 and a x b becomes the zero vector -- the standard test for two nonzero vectors being parallel, mirroring how a.b = 0 tests for perpendicularity. In component form, for a = a1 i+a2 j+a3 k and b = b1 i+b2 j+b3 k, the cross product works out to (a2 b3 - a3 b2) i - (a1 b3 - a3 b1) j + (a1 b2 - a2 b1) k, organised as a determinant with i, j, k on top and the two vectors' components below. Since |a x b| = |a||b| sin(theta) is the parallelogram area for sides a and b, the cross product also works as an area tool: a parallelogram with adjacent sides a and b has area |a x b|, and a triangle with two sides a and b has area (1/2)|a x b|.

Stepping back, the dot product and the cross product answer two genuinely different geometric questions, and confusing them is one of the most common errors in this topic. The dot product collapses two vectors down into a single number that measures alignment, how much one vector points along the other, and it vanishes exactly when the vectors are perpendicular. The cross product instead builds an entirely new vector that measures a turning or spanning effect, capturing both an area (its magnitude) and an orientation (its direction, via the right-hand rule), and it vanishes exactly when the vectors are parallel. This split mirrors a genuine physical distinction: the work done by a force moving an object is a dot product (force . displacement, a plain number of joules), while the turning effect of a force about a pivot, its torque, is a cross product (a vector, since a torque itself has a direction, the axis it turns around). Together with vector addition, scalar multiplication, and the section formula from earlier in this chapter, these two products complete the basic algebraic toolkit for vectors: every vector calculation from here on, whether checking perpendicularity, computing an angle, finding an area, or building a vector perpendicular to a given plane, reduces to some combination of these operations applied to components.