University Physics Volume 1

# Key Equations

### Key Equations

 Multiplication by a scalar (vector equation) $B→=αA→B→=αA→$ Multiplication by a scalar (scalar equation for magnitudes) $B=|α|AB=|α|A$ Resultant of two vectors $D→AD=D→AC+D→CDD→AD=D→AC+D→CD$ Commutative law $A→+B→=B→+A→A→+B→=B→+A→$ Associative law $(A→+B→)+C→=A→+(B→+C→)(A→+B→)+C→=A→+(B→+C→)$ Distributive law $α1A→+α2A→=(α1+α2)A→α1A→+α2A→=(α1+α2)A→$ The component form of a vector in two dimensions $A→=Axi^+Ayj^A→=Axi^+Ayj^$ Scalar components of a vector in two dimensions ${Ax=xe−xb Ay=ye−yb{Ax=xe−xb Ay=ye−yb$ 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 $Az=ze−zbAz=ze−zb$ Magnitude of a vector in three dimensions $A=Ax2+Ay2+Az2A=Ax2+Ay2+Az2$ Distributive property $α(A→+B→)=αA→+αB→α(A→+B→)=αA→+αB→$ Antiparallel vector to $A→A→$ $−A→=−Axi^−Ayj^−Azk^−A→=−Axi^−Ayj^−Azk^$ 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 $V^=V→VV^=V→V$ Definition of the scalar product $A→·B→=ABcosφA→·B→=ABcosφ$ Commutative property of the scalar product $A→·B→=B→·A→A→·B→=B→·A→$ Distributive property of the scalar product $A→·(B→+C→)=A→·B→+A→·C→A→·(B→+C→)=A→·B→+A→·C→$ 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 $cosφ=A→·B→ABcosφ=A→·B→AB$ 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 $A→×B→=−B→×A→A→×B→=−B→×A→$ Distributive property of the vector product $A→×(B→+C→)=A→×B→+A→×C→A→×(B→+C→)=A→×B→+A→×C→$ 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 scalarcomponents of vectors $A→×B→=(AyBz−AzBy)i^+(AzBx−AxBz)j^+(AxBy−AyBx)k^A→×B→=(AyBz−AzBy)i^+(AzBx−AxBz)j^+(AxBy−AyBx)k^$
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