Skip to ContentGo to accessibility pageKeyboard shortcuts menu
OpenStax Logo

Key Equations

Multiplication by a scalar (vector equation) B=αAB=αA
Multiplication by a scalar (scalar equation for magnitudes) B=|α|AB=|α|A
Resultant of two vectors DAD=DAC+DCDDAD=DAC+DCD
Commutative law A+B=B+AA+B=B+A
Associative law (A+B)+C=A+(B+C)(A+B)+C=A+(B+C)
Distributive law α1A+α2A=(α1+α2)Aα1A+α2A=(α1+α2)A
The component form of a vector in two dimensions A=Axi^+Ayj^A=Axi^+Ayj^
Scalar components of a vector in two dimensions {Ax=xexb Ay=yeyb{Ax=xexb Ay=yeyb
Magnitude of a vector in a plane A=Ax2+Ay2A=Ax2+Ay2
The direction angle of a vector in a plane θA=tan−1(AyAx)θA=tan−1(AyAx)
Scalar components of a vector in a plane {Ax=AcosθA Ay=AsinθA{Ax=AcosθA Ay=AsinθA
Polar coordinates in a plane {x=rcosφ y=rsinφ{x=rcosφ y=rsinφ
The component form of a vector in three dimensions A=Axi^+Ayj^+Azk^A=Axi^+Ayj^+Azk^
The scalar z-component of a vector in three dimensions Az=zezbAz=zezb
Magnitude of a vector in three dimensions A=Ax2+Ay2+Az2A=Ax2+Ay2+Az2
Distributive property α(A+B)=αA+αBα(A+B)=αA+αB
Antiparallel vector to AA A=Axi^Ayj^Azk^A=Axi^Ayj^Azk^
Equal vectors A=B{Ax=Bx Ay=By Az=BzA=B{Ax=Bx Ay=By Az=Bz
Components of the resultant of N vectors {FRx=k=1NFkx=F1x+F2x++FNx FRy=k=1NFky=F1y+F2y++FNy FRz=k=1NFkz=F1z+F2z++FNz{FRx=k=1NFkx=F1x+F2x++FNx FRy=k=1NFky=F1y+F2y++FNy FRz=k=1NFkz=F1z+F2z++FNz
General unit vector V^=VVV^=VV
Definition of the scalar product A·B=ABcosφA·B=ABcosφ
Commutative property of the scalar product A·B=B·AA·B=B·A
Distributive property of the scalar product A·(B+C)=A·B+A·CA·(B+C)=A·B+A·C
Scalar product in terms of scalar components of vectors A·B=AxBx+AyBy+AzBzA·B=AxBx+AyBy+AzBz
Cosine of the angle between two vectors cosφ=A·BABcosφ=A·BAB
Dot products of unit vectors i^·j^=j^·k^=k^·i^=0i^·j^=j^·k^=k^·i^=0
Magnitude of the vector product (definition) |A×B|=ABsinφ|A×B|=ABsinφ
Anticommutative property of the vector product A×B=B×AA×B=B×A
Distributive property of the vector product A×(B+C)=A×B+A×CA×(B+C)=A×B+A×C
Cross products of unit vectors {i^×j^=+k^, j^×k^=+i^, k^×i^=+j^.{i^×j^=+k^, j^×k^=+i^, k^×i^=+j^.
The cross product in terms of scalar
components of vectors
A×B=(AyBzAzBy)i^+(AzBxAxBz)j^+(AxByAyBx)k^A×B=(AyBzAzBy)i^+(AzBxAxBz)j^+(AxByAyBx)k^
Order a print copy

As an Amazon Associate we earn from qualifying purchases.

Citation/Attribution

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Attribution information
  • If you are redistributing all or part of this book in a print format, then you must include on every physical page the following attribution:
    Access for free at https://openstax.org/books/university-physics-volume-1/pages/1-introduction
  • If you are redistributing all or part of this book in a digital format, then you must include on every digital page view the following attribution:
    Access for free at https://openstax.org/books/university-physics-volume-1/pages/1-introduction
Citation information

© Jan 19, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.