Key relationship
A · B = AB cos θ = AₓBₓ + AᵧBᵧ; (A × B)z = AB sin θ = AₓBᵧ − AᵧBₓ.
Physics · Class 11
Dot & cross products
The dot product is a scalar measuring alignment: positive for acute angles, zero at 90°, negative for obtuse angles. The cross product is a vector perpendicular to both inputs. Its z-component is signed in this plane. The right-hand rule gives its direction, and reversing the order reverses that direction.
Open the interactive experiment →A product can measure alignment or turning.
புள்ளிப் பெருக்கல் ஒரு திசையிலி; திசையன்களின் ஒத்த திசையை அளக்கிறது. குறுக்குப் பெருக்கல் இரு திசையன்களுக்கும் செங்குத்தான திசையன். வரிசையை மாற்றினால் அதன் திசை மாறும்.
Key notes, relationships and model limits
Remember
The dot product is a scalar measuring alignment: positive for acute angles, zero at 90°, negative for obtuse angles. The cross product is a vector perpendicular to both inputs. Its z-component is signed in this plane. The right-hand rule gives its direction, and reversing the order reverses that direction.
Common trap
The cross-product magnitude is nonnegative, but its z-component can be negative. A × B = −(B × A).
Model limits
A and B are planar vectors in arbitrary units, with A along +x. Positive angles turn counterclockwise from A to B. +z points out of the page. Product units are the product of the input units; work and torque require appropriate physical input quantities.
What you will learn
Calculate dot and cross products; interpret angle signs; apply the right-hand rule.
Try the experiment
Compare parallel and perpendicular presets. Switch from +90° to −90° and observe the cross direction.
Think of it this way
Pushing along motion transfers energy; pushing across a lever can turn it. Dot and cross products describe these different geometric relationships.
Worked example
For A=5î and B at 60° with magnitude4: A·B=5×4×cos60°=10; (A×B)z=5×4×sin60°≈17.32.
Why this matters
Work uses force · displacement, while torque uses position × force. The operations answer different physical questions.
Before you start
Rectangular axes · Pythagoras · sine and cosine
Check your understanding
A points right and B points up. Which way does A × B point?
- Into the page
- Out of the page
- Along A
Read the explanation
î × ĵ = k̂. With x right and y up, +z is out of the page. Reversing the order would point into the page.
Course connection: CBSE XI Motion in a Plane; NEET/JEE kinematics. Use the assumptions stated in each question.