Some quantities need only a number: 5 seconds, 30 km/h, 10 grams. Others also need a direction: a push of 10 newtons which way? a walk of 4 km towards where? These are vectors, and we draw them as arrows. In this chapter you learn to add arrows, stretch them, split them into i, j and k parts, and multiply them in two different ways: the dot product measures how much two arrows agree, and the cross product builds a new arrow standing straight up from both.
Some quantities are just a size. A vector is a size together with a direction, so we draw it as an arrow.
A scalar has a magnitude only (mass, time, speed, temperature). A vector has a magnitude and a direction (displacement, velocity, force). Draw it as a directed line segment from the initial point A to the terminal point B. Its length |AB⃗| is the magnitude.
The arrow OP⃗ from the origin O to the point P(x, y, z) is the position vector of P. This is its magnitude.
OP⃗ makes angles α, β, γ with the positive x, y and z axes (its direction angles). Their cosines l = cos α, m = cos β, n = cos γ are the direction cosines. Here r = |OP⃗|.
Any numbers proportional to the direction cosines (for example x, y, z themselves) are direction ratios.
Draw a displacement of 40 km, 30° west of south.
Scalars or vectors: 5 s, 1000 cm³, 10 N, 30 km/h, 10 g/cm³, 20 m/s north.
Some vectors come up so often that they have their own names. Each name says what is special about one arrow, or what two arrows share.
| Name | What it means |
|---|---|
| Zero vector 0⃗ | length 0 and no fixed direction: its start and end are the same point |
| Unit vector â | length 1; â is the unit vector along a⃗ |
| Coinitial AB⃗, AC⃗ | start at the same point |
| Collinear a⃗ ∥ b⃗ | lie along parallel lines (also called parallel), whatever their lengths and senses |
| Equal a⃗ = b⃗ | same length and same direction, wherever they start |
| Negative BA⃗ = −AB⃗ | same length, opposite direction |
In Fig 10.5, which vectors are collinear, equal, coinitial?
To add two arrows, walk along one and then the other. The sum is the single arrow from where you started to where you ended.
Put the tail of b⃗ at the head of a⃗: the sum a⃗ + b⃗ goes from the tail of a⃗ to the head of b⃗. Going round a closed triangle brings you back to the start.
Put both arrows at the same start. The sum is the diagonal of the parallelogram they span: the same arrow as the triangle law gives.
Subtraction adds the negative.
Rules for adding
Addition is commutative and associative, and 0⃗ is the additive identity.
Split any arrow into three steps along the axes. Then working with vectors becomes working with ordinary numbers.
i, j, k are the unit vectors along the x, y and z axes. For the position vector of P(x, y, z), x, y, z are the scalar components and xi, yj, zk the vector components.
Multiplying by a scalar λ stretches the length by |λ|, and reverses the direction if λ < 0.
Divide a vector by its own length: the result is the unit vector along a⃗.
The vector from P₁(x₁, y₁, z₁) to P₂(x₂, y₂, z₂).
| Vectors | In components |
|---|---|
| a⃗ + b⃗ | (a₁ + b₁)i + (a₂ + b₂)j + (a₃ + b₃)k |
| λa⃗ | (λa₁)i + (λa₂)j + (λa₃)k |
| a⃗ = b⃗ | exactly when a₁ = b₁, a₂ = b₂, a₃ = b₃ |
| a⃗, b⃗ collinear | exactly when b⃗ = λa⃗, that is b₁a₁ = b₂a₂ = b₃a₃ |
xi + 2j + zk = 2i + yj + k.
a⃗ = i + 2j, b⃗ = 2i + j: equal lengths? equal vectors?
Unit vector along 2i + 3j + k.
Length 7 along i − 2j.
Unit vector along (2i + 2j − 5k) + (2i + j + 3k).
Direction ratios and cosines of i + j − 2k.
Vector from P(2, 3, 0) to Q(−1, −2, −4).
A point that divides PQ in a ratio is a weighted average of the two ends. It is pulled towards the end with the bigger weight.
P and Q have position vectors a⃗ and b⃗. R lies between P and Q and divides PQ internally: PR : RQ = m : n.
R lies outside the segment and divides PQ externally: PR : QR = m : n.
The midpoint is the case m = n.
The ratio number next to Q multiplies Q’s vector, so the point is pulled towards the end with the bigger weight.
OP⃗ = 3a⃗ − 2b⃗, OQ⃗ = a⃗ + b⃗; divide in 2 : 1.
A(2i − j + k), B(i − 3j − 5k), C(3i − 4j − 4k) form a right triangle.
The dot product turns two arrows into one number. It measures how much they point the same way, and it is 0 at a right angle.
θ is the angle between them, 0 ≤ θ ≤ π. The answer is a number: the scalar (dot) product. It is 0 if either vector is 0⃗.
The dot product is commutative and distributes over addition. With i · i = j · j = k · k = 1 and i · j = j · k = k · i = 0, that gives this rule. Also a⃗ · a⃗ = |a⃗|².
Positive for an acute angle, 0 for a right angle, negative for an obtuse angle. So a⃗ ⊥ b⃗ exactly when a⃗ · b⃗ = 0.
The projection of a⃗ on b⃗: the length of a⃗’s shadow on the line of b⃗. The projection vector is (a⃗ · b̂) b̂.
|a⃗| = 1, |b⃗| = 2, a⃗ · b⃗ = 1: the angle.
Angle between i + j − k and i − j + k.
(a⃗ + b⃗) ⊥ (a⃗ − b⃗) for a⃗ = 5i − j − 3k, b⃗ = i + 3j − 5k.
Projection of 2i + 3j + 2k on i + 2j + k.
|a⃗| = 2, |b⃗| = 3, a⃗ · b⃗ = 4: |a⃗ − b⃗|.
â unit, (x⃗ − â) · (x⃗ + â) = 8.
|a⃗ · b⃗| ≤ |a⃗||b⃗|.
|a⃗ + b⃗| ≤ |a⃗| + |b⃗|.
A(−2i + 3j + 5k), B(i + 2j + 3k), C(7i − k) collinear.
The cross product turns two arrows into a new arrow that stands straight up from both. Its length is the area they span.
For non-zero, non-parallel a⃗ and b⃗: the vector (cross) product. n̂ is the unit vector perpendicular to both, chosen by the right-hand rule: curl the fingers from a⃗ to b⃗, and the thumb points along n̂.
This is the area of the parallelogram with sides a⃗ and b⃗. A triangle with sides a⃗ and b⃗ has area ½|a⃗ × b⃗|.
The cross product is not commutative. And if a⃗ ∥ b⃗, then a⃗ × b⃗ = 0⃗.
The cross product is this determinant, with i, j, k in the top row.
The unit vectors
|a⃗ × b⃗| for 2i + j + 3k and 3i + 5j − 2k.
Unit vector ⊥ to a⃗ + b⃗ and a⃗ − b⃗.
Triangle A(1, 1, 1), B(1, 2, 3), C(2, 3, 1).
Parallelogram on 3i + j + 4k and i − j + k.
All unit vectors in the XY-plane.
Angle between AB⃗ and CD⃗; collinear?
|a⃗| = 3, |b⃗| = 4, |c⃗| = 5, each ⊥ the sum of the other two.
a⃗ + b⃗ + c⃗ = 0⃗, lengths 3, 4, 2: a⃗ · b⃗ + b⃗ · c⃗ + c⃗ · a⃗.
Split β⃗ = 2i + j − 3k along and across α⃗ = 3i − j.
Fifteen quick questions to check yourself. Tap an option to see whether it is right, and why.
Every MCQ from Exercises 10.2–10.4 and the miscellaneous exercise, plus six concept checks.
A vector has a magnitude and a direction. OP⃗ = xi + yj + zk and |OP⃗| = √(x² + y² + z²).
Direction cosines: l = x/r, m = y/r, n = z/r, with l² + m² + n² = 1.
Add head to tail (triangle law) or along the diagonal (parallelogram law); in components, add matching parts. λa⃗ stretches by |λ|; â = a⃗/|a⃗|.
AB⃗ = (position of B) − (position of A). Section formula: (mb⃗ + na⃗)/(m + n) internally, (mb⃗ − na⃗)/(m − n) externally.
a⃗ · b⃗ = |a⃗||b⃗| cos θ = a₁b₁ + a₂b₂ + a₃b₃, a number. Zero means perpendicular. Projection of a⃗ on b⃗ = a⃗ · b⃗/|b⃗|.
a⃗ × b⃗ = |a⃗||b⃗| sin θ n̂: a vector perpendicular to both, found by the i, j, k determinant.
|a⃗ × b⃗| is the parallelogram area, and b⃗ × a⃗ = −a⃗ × b⃗.
Already knew that the combined action of two forces could be found by the parallelogram law.
Analysed the geometric addition of forces in De Beghinselen der Weeghconst ("Principles of the Art of Weighing").
Gave the complex number a + ib a picture as a directed line segment in a plane.
First used the word vector (from the Latin vectus, "to carry") in Lectures on Quaternions.
Separated the scalar and vector parts of the quaternion product and created the vector analysis used today.
A vector is an arrow: it has a length and a direction. Each walkthrough draws the arrows, in the plane or in 3D space with the x, y and z axes, and builds the answer step by step: adding head to tail, stretching by a number, dropping a shadow for the dot product, spanning a parallelogram for the cross product. Press Next step (or Play); each step is explained in detail under the controls, with links to the basics it uses.
Draw a displacement, sort scalars from vectors, and name the types of vectors.
Every vector in space is xi + yj + zk. Equal vectors, lengths, unit vectors, direction cosines and the vector between two points.
The point dividing PQ in the ratio m : n, inside or outside; and Pythagoras with vectors.
a⃗ · b⃗ = |a⃗||b⃗| cos θ: angles, perpendicular vectors, projections, and two famous inequalities.
a⃗ × b⃗ is perpendicular to both, and its length is the area of the parallelogram they span.
Unit vectors in a plane, collinear vectors, a closed triangle and resolving a vector into two parts.
A boat crossing a flowing river, two ropes hauling a load, a trolley pushed at an angle, a spanner turning a bolt, and a walk across a field: in each one the answer depends on directions as well as sizes. Adding vectors gives the combined effect, the dot product gives work, and the cross product gives turning.
A boat moves at 4 m/s through the water; the river flows at 2 m/s. The boat's real velocity is the vector sum of its own velocity and the current's. Point it straight across and it drifts downstream; aim it upstream at the right angle (sin φ = 2/4, φ = 30°) and the sum points straight across.
Two people haul a crate with ropes, each pulling with 300 N. The crate feels one force: the diagonal of the parallelogram of the two pulls. The wider apart the ropes, the smaller the resultant: |R⃗|² = |F⃗₁|² + |F⃗₂|² + 2 F⃗₁ · F⃗₂.
You push a trolley with 50 N along a handle tilted θ below the horizontal, and it rolls 8 m. The work done is the dot product W = F⃗ · d⃗ = |F⃗||d⃗| cos θ: only the shadow of the force on the path counts. The downward part just presses the trolley into the floor.
The turning effect of a force on a bolt is the torque τ⃗ = r⃗ × F⃗, where r⃗ runs from the bolt to your hand. Its size |r⃗||F⃗| sin θ is biggest when you pull at 90° to the handle and far from the bolt; its direction (along the bolt, by the right-hand rule) says whether the bolt tightens or loosens.
A girl walks 4 km west, then 3 km in a direction 30° east of north. Her displacement is the sum of the two legs: −4i + (3 sin 30°)i + (3 cos 30°)j = −(5/2)i + (3√3/2)j, about 3.61 km from the start. Change the bearing of the second leg and watch the straight-line distance home.
Decide first, then tap each card.
Every question that asks you to find, show, write or classify has an animated walkthrough here; the 9 multiple-choice questions are in the Lesson’s self-check. Pick a question from the chips above each player, then press Play or step through with Next. Each step draws the arrows it talks about, in the plane or in 3D, and ends with the answer on the board.
Draw a displacement, sort quantities into scalars and vectors, and name coinitial, equal and collinear vectors.
Magnitudes, equal and parallel vectors, unit vectors, direction cosines, the section formula and a right triangle. (Q18–Q19 are in the quiz.)
Angles, projections, perpendicular vectors, expanding dot products and a right triangle. (Q18 is in the quiz.)
Determinants, perpendicular unit vectors, parallel vectors, and the areas of triangles and parallelograms. (Q11–Q12 are in the quiz.)
A walk with bearings, collinear points, external division, a parallelogram, and a vector fixed by three conditions. (Q16–Q19 are in the quiz.)
Vector algebra rests on eight older ideas. Here is each one from its roots: why some quantities need a direction, how three numbers fix a point in space, Pythagoras used twice, the cosine as a shadow, compass bearings, ratios and weighted averages, expanding brackets, and the determinant as an area. Every example and exercise links here exactly where you need it, and a button takes you straight back to the problem.
If you walk 3 km and then 4 km, how far are you from home? It depends on the directions. The distance walked is always 7 km, a plain number (a scalar). But the displacement, where you ended up compared with where you began, can be anything from 1 km to 7 km, and it points somewhere. A quantity like that, which needs a size and a direction, is a vector. That is why we draw vectors as arrows: the length shows the size, the arrowhead shows the direction. Force, velocity, acceleration and displacement are vectors; mass, time, speed, temperature, energy and distance are scalars.
In a plane, two perpendicular number lines (the x and y axes) let two numbers (x, y) name any point. Space needs a third direction, straight up out of the plane: the z-axis, perpendicular to both. The axes are arranged right-handed: curl the fingers of your right hand from x towards y and your thumb points along z. The point P(x, y, z) is reached from the origin O by walking x along the x-axis, then y parallel to the y-axis, then z parallel to the z-axis. The three coordinate planes (XY, YZ, ZX) split space into eight octants, like the four quadrants of the plane. On a flat page we draw the x-axis slanting towards us, so the picture is a projection: equal lengths along x look shorter.
In a right triangle, hypotenuse² = (one side)² + (other side)². In space, use it twice: across the floor, then straight up.
The floor diagonal from O to (x, y, 0) has this length: one right triangle.
Then go straight up by z. The vertical edge is perpendicular to the floor, so a second right triangle gives the distance from O to P(x, y, z).
Use the differences, and PQ is the square root of this. Signs do not matter because every term is squared.
Put an arrow of length 1 at angle θ to a line and shine a light straight down onto the line. The shadow has length cos θ; the height of the tip is sin θ.
Past 90° the shadow falls behind the start, so cos θ is negative; sin θ stays positive for 0 < θ < π.
Pythagoras on the unit circle.
That one picture explains most of this chapter: direction cosines are the shadows of a unit vector on the three axes, the dot product |a⃗||b⃗| cos θ is length × shadow, and the cross product |a⃗||b⃗| sin θ is base × height, the area of a parallelogram.
A compass direction like 30° east of north is read in three parts: start facing north, turn 30°, towards east. So "30° west of south" starts facing south and turns 30° towards west. North-west is exactly halfway between north and west (45°). To turn a bearing into components, take i pointing east and j pointing north. A displacement r at θ east of north has east part r sin θ and north part r cos θ: the cosine goes with the direction you started from.
“R divides PQ in the ratio 2 : 1” means PR is 2 parts and RQ is 1 part: R is 2/3 of the way from P to Q.
On a seesaw from P to Q put weight n at P and weight m at Q. It balances at the point R with this ratio, because the heavier weight must sit nearer the pivot.
This weighted average is the section formula, and it is why the numbers “cross over”.
The other use of ratios: a⃗ and b⃗ are collinear when this holds (all components scaled by the same number λ).
A square of side x + y splits into a square x², a square y² and two rectangles xy. The dot product follows the same algebra.
With numbers
This works because the dot product distributes and is commutative. Likewise (a⃗ + b⃗) · (a⃗ − b⃗) = |a⃗|² − |b⃗|².
The cross product also distributes, but it is not commutative, so keep the order when you expand it.
A determinant measures the signed area of the parallelogram that two vectors span. That is why it can build the cross product.
From Chapter 4. For the vectors (a, b) and (c, d), ad − bc is the area of the parallelogram they span, with a sign: positive if the turn from the first to the second is anticlockwise, negative if clockwise.
Expand along the first row. Note the minus in the middle.
Putting i, j, k in the top row gives exactly a⃗ × b⃗: each component is a 2 × 2 area of the shadow of the parallelogram on one coordinate plane.
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