Intuition The ONE core ideaA force acting away from a pivot doesn't just push — it twists , and the strength of that twist is called torque. Everything on the parent page is built from three raw ingredients: an arrow that says where (r arrow that says how you push (F machine that combines two arrows into a twist (the cross product).
This page assumes you know nothing . Every squiggle on the parent page is unpacked here, one at a time, each one leaning on the one before it. By the end you will have earned every symbol in τ = r × F
Definition A vector — the "arrow" quantityA vector is a quantity that needs both a size and a direction to be complete. We draw it as an arrow : the length is the size, the way it points is the direction. We write it with a little arrow on top, like r F
Intuition Why we need arrows at all"5 metres" tells you a distance but not which way . "5 metres to the right" is complete — that is a vector. A push has the same problem: "50 newtons" is useless until you say which direction you push. Rotation is all about direction, so we cannot escape arrows.
Contrast this with a scalar — a plain number with no direction , like temperature (2 0 ∘ 3 kg
Before anything can twist, we need a fixed point to twist around . Call it the pivot (also "axis" or "hinge"). Think of the exact line the door swings on.
Definition Position vectorr r starts at the pivot and ends at the point where the force is applied . Its length r
Common mistake The classicr Feels right: "r The flaw: an arrow needs a tail and a head — the tail is the pivot , the head is the contact point. Start it anywhere else and every later number is wrong.
Look at figure s02: the tail of the blue arrow sits on the hinge, the head sits on the handle. The far-off handle gives a long r short r
F F F newtons , symbol N), its direction is the way you push. One newton is roughly the weight of a small apple in your hand.
Now the crucial move. A push can point any way relative to r θ
θ θ r F θ = 0 ∘ straight out along r θ = 9 0 ∘ exactly sideways to r θ = 18 0 ∘ straight back toward the pivot.
We now ask: how much of the push actually goes "sideways"? To answer we split F r
Intuition Why split the force at allOnly the sideways part of a push can carry the contact point around the pivot. The part pointing along r
To split an arrow into "along" and "across" pieces we use a right triangle — and right triangles are exactly what sin cos
Definition Sine and cosine on a right triangleTake a right triangle with one angle θ cos θ = longest side side next to θ , sin θ = longest side side opposite θ . F F cos θ F sin θ
In figure s03 the pink arrow (F ⊥ = F sin θ F ∥ = F cos θ
θ cos Feels right: we meet cos The flaw: here we need the piece across r F sin θ θ = 0 ∘ sin 0 ∘ = 0 Sin to spin.
Now the star symbol of the whole topic: the × r × F
a × b cross product is an operation that eats two vectors and produces a new vector . Its size is
∣ a × b ∣ = a b sin θ , perpendicular to both input arrows.
sin θ area of the parallelogram you can build from the two arrows. Base = a = b sin θ = ab sin θ θ = 0 exactly the door pushed into its hinges.
This is why the parent page's τ = r F sin θ forced by the cross product's built-in sin θ Cross Product (Vector Algebra) to go deeper on this machine itself.
Definition Right-hand rule — reading the twist directionPoint your right hand's fingers along r F thumb points along τ out of the page (we call this + into the page (−
τ τ twist . Its length τ = r F sin θ Unit: newton-metre (N·m) — a distance (m) multiplied by a force (N).
Common mistake "N·m means torque is energy."Feels right: energy (the joule) is also N·m. The flaw: torque is a vector with an axis; energy is a plain scalar. Same units, different creatures — never add them. Write N·m for torque, J for energy.
The parent page writes forces as ( x , y , z ) 6 k ^
i ^ , j ^ , k ^ exactly 1 , pointing along the three axes: i ^ j ^ k ^ ( 2 , 0 , 0 ) = 2 i ^ ^
Vector: size plus direction
Position vector r from pivot
Angle theta between r and F
Split force: sin theta sideways
Direction by right hand rule
Each arrow means "you need the top box before the bottom box makes sense." Notice that everything flows down into τ up from the single idea of a vector.
Return to the parent whenever you want the physics story: Torque — definition & physical meaning .
Test yourself — cover the right side and answer before revealing.
A vector differs from a scalar because a vector carries a direction as well as a size; a scalar is just a number.
Where does the arrow r it starts at the pivot (axis) and ends at the point where the force is applied.
What does θ the angle between
r and
F when their tails are joined.
Which force component causes rotation, and what is it? the perpendicular (sideways) component, F sin θ
Why does sin θ cos θ only the sideways part twists; at
θ = 0 ∘ the push is along
r and
sin 0 ∘ = 0 gives zero torque, correctly.
What does the cross product × a new vector perpendicular to both inputs, with size ab sin θ
What does a hat, as in k ^ a unit vector — an arrow of length exactly 1 marking a pure direction (here, out of the page).
How do you find the direction of τ right-hand rule — curl fingers from
r to
F , thumb points along
τ .
The 2D coordinate formula for torque is τ z = x F y − y F x
Torque's unit and why it isn't energy newton-metre (N·m); same units as the joule but torque is a vector, energy is a scalar.
