{"id":1390,"date":"2018-02-06T15:31:23","date_gmt":"2018-02-06T15:31:23","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/?post_type=chapter&p=1390"},"modified":"2018-02-06T15:31:23","modified_gmt":"2018-02-06T15:31:23","slug":"2-chapter-review","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-osuniversityphysics\/chapter\/2-chapter-review\/","title":{"raw":"2 Chapter Review","rendered":"2 Chapter Review"},"content":{"raw":"
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Key Terms<\/span><\/h3>\r\n
\r\n \t
anticommutative property<\/strong><\/dt>\r\n \t
change in the order of operation introduces the minus sign<\/dd>\r\n<\/dl>\r\n
\r\n \t
antiparallel vectors<\/strong><\/dt>\r\n \t
two vectors with directions that differ by\u00a0180<\/span>\u00b0<\/span><\/span><\/span><\/span><\/span><\/span>180\u00b0<\/span><\/span><\/dd>\r\n<\/dl>\r\n
\r\n \t
associative<\/strong><\/dt>\r\n \t
terms can be grouped in any fashion<\/dd>\r\n<\/dl>\r\n
\r\n \t
commutative<\/strong><\/dt>\r\n \t
operations can be performed in any order<\/dd>\r\n<\/dl>\r\n
\r\n \t
component form of a vector<\/strong><\/dt>\r\n \t
a vector written as the vector sum of its components in terms of unit vectors<\/dd>\r\n<\/dl>\r\n
\r\n \t
corkscrew right-hand rule<\/strong><\/dt>\r\n \t
a rule used to determine the direction of the vector product<\/dd>\r\n<\/dl>\r\n
\r\n \t
cross product<\/strong><\/dt>\r\n \t
the result of the vector multiplication of vectors is a vector called a cross product; also called a vector product<\/dd>\r\n<\/dl>\r\n
\r\n \t
difference of two vectors<\/strong><\/dt>\r\n \t
vector sum of the first vector with the vector antiparallel to the second<\/dd>\r\n<\/dl>\r\n
\r\n \t
direction angle<\/strong><\/dt>\r\n \t
in a plane, an angle between the positive direction of the\u00a0x<\/em>-axis and the vector, measured counterclockwise from the axis to the vector<\/dd>\r\n<\/dl>\r\n
\r\n \t
displacement<\/strong><\/dt>\r\n \t
change in position<\/dd>\r\n<\/dl>\r\n
\r\n \t
distributive<\/strong><\/dt>\r\n \t
multiplication can be distributed over terms in summation<\/dd>\r\n<\/dl>\r\n
\r\n \t
dot product<\/strong><\/dt>\r\n \t
the result of the scalar multiplication of two vectors is a scalar called a dot product; also called a scalar product<\/dd>\r\n<\/dl>\r\n
\r\n \t
equal vectors<\/strong><\/dt>\r\n \t
two vectors are equal if and only if all their corresponding components are equal; alternately, two parallel vectors of equal magnitudes<\/dd>\r\n<\/dl>\r\n
\r\n \t
magnitude<\/strong><\/dt>\r\n \t
length of a vector<\/dd>\r\n<\/dl>\r\n
\r\n \t
null vector<\/strong><\/dt>\r\n \t
a vector with all its components equal to zero<\/dd>\r\n<\/dl>\r\n
\r\n \t
orthogonal vectors<\/strong><\/dt>\r\n \t
two vectors with directions that differ by exactly\u00a090<\/span>\u00b0<\/span><\/span><\/span><\/span><\/span><\/span>90\u00b0<\/span><\/span>, synonymous with perpendicular vectors<\/dd>\r\n<\/dl>\r\n
\r\n \t
parallel vectors<\/strong><\/dt>\r\n \t
two vectors with exactly the same direction angles<\/dd>\r\n<\/dl>\r\n
\r\n \t
parallelogram rule<\/strong><\/dt>\r\n \t
geometric construction of the vector sum in a plane<\/dd>\r\n<\/dl>\r\n
\r\n \t
polar coordinate system<\/strong><\/dt>\r\n \t
an orthogonal coordinate system where location in a plane is given by polar coordinates<\/dd>\r\n<\/dl>\r\n
\r\n \t
polar coordinates<\/strong><\/dt>\r\n \t
a radial coordinate and an angle<\/dd>\r\n<\/dl>\r\n
\r\n \t
radial coordinate<\/strong><\/dt>\r\n \t
distance to the origin in a polar coordinate system<\/dd>\r\n<\/dl>\r\n
\r\n \t
resultant vector<\/strong><\/dt>\r\n \t
vector sum of two (or more) vectors<\/dd>\r\n<\/dl>\r\n
\r\n \t
scalar<\/strong><\/dt>\r\n \t
a number, synonymous with a scalar quantity in physics<\/dd>\r\n<\/dl>\r\n
\r\n \t
scalar component<\/strong><\/dt>\r\n \t
a number that multiplies a unit vector in a vector component of a vector<\/dd>\r\n<\/dl>\r\n
\r\n \t
scalar equation<\/strong><\/dt>\r\n \t
equation in which the left-hand and right-hand sides are numbers<\/dd>\r\n<\/dl>\r\n
\r\n \t
scalar product<\/strong><\/dt>\r\n \t
the result of the scalar multiplication of two vectors is a scalar called a scalar product; also called a dot product<\/dd>\r\n<\/dl>\r\n
\r\n \t
scalar quantity<\/strong><\/dt>\r\n \t
quantity that can be specified completely by a single number with an appropriate physical unit<\/dd>\r\n<\/dl>\r\n
\r\n \t
tail-to-head geometric construction<\/strong><\/dt>\r\n \t
geometric construction for drawing the resultant vector of many vectors<\/dd>\r\n<\/dl>\r\n
\r\n \t
unit vector<\/strong><\/dt>\r\n \t
vector of a unit magnitude that specifies direction; has no physical unit<\/dd>\r\n<\/dl>\r\n
\r\n \t
unit vectors of the axes<\/strong><\/dt>\r\n \t
unit vectors that define orthogonal directions in a plane or in space<\/dd>\r\n<\/dl>\r\n
\r\n \t
vector<\/strong><\/dt>\r\n \t
mathematical object with magnitude and direction<\/dd>\r\n<\/dl>\r\n
\r\n \t
vector components<\/strong><\/dt>\r\n \t
orthogonal components of a vector; a vector is the vector sum of its vector components.<\/dd>\r\n<\/dl>\r\n
\r\n \t
vector equation<\/strong><\/dt>\r\n \t
equation in which the left-hand and right-hand sides are vectors<\/dd>\r\n<\/dl>\r\n
\r\n \t
vector product<\/strong><\/dt>\r\n \t
the result of the vector multiplication of vectors is a vector called a vector product; also called a cross product<\/dd>\r\n<\/dl>\r\n
\r\n \t
vector quantity<\/strong><\/dt>\r\n \t
physical quantity described by a mathematical vector\u2014that is, by specifying both its magnitude and its direction; synonymous with a vector in physics<\/dd>\r\n<\/dl>\r\n
\r\n \t
vector sum<\/strong><\/dt>\r\n \t
resultant of the combination of two (or more) vectors<\/dd>\r\n<\/dl>\r\n<\/div>\r\n<\/div>\r\n
\r\n
\r\n

Key Equations<\/span><\/h3>\r\n
\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Multiplication by a scalar (vector equation)<\/td>\r\nB<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>\u03b1<\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>B\u2192=\u03b1A\u2192<\/span><\/span><\/td>\r\n<\/tr>\r\n
Multiplication by a scalar (scalar equation for magnitudes)<\/td>\r\nB<\/span>=<\/span>|<\/span>\u03b1<\/span>|<\/span>A<\/span><\/span><\/span><\/span><\/span><\/span>B=|\u03b1|A<\/span><\/span><\/td>\r\n<\/tr>\r\n
Resultant of two vectors<\/td>\r\nD<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>A<\/span>D<\/span><\/span><\/span>=<\/span>D<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>A<\/span>C<\/span><\/span><\/span>+<\/span>D<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>C<\/span>D<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>D\u2192AD=D\u2192AC+D\u2192CD<\/span><\/span><\/td>\r\n<\/tr>\r\n
Commutative law<\/td>\r\nA<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>+<\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>+<\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A\u2192+B\u2192=B\u2192+A\u2192<\/span><\/span><\/td>\r\n<\/tr>\r\n
Associative law<\/td>\r\n(<\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>+<\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>)<\/span>+<\/span>C<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>+<\/span>(<\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>+<\/span>C<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span>(A\u2192+B\u2192)+C\u2192=A\u2192+(B\u2192+C\u2192)<\/span><\/span><\/td>\r\n<\/tr>\r\n
Distributive law<\/td>\r\n\u03b1<\/span>1<\/span><\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>+<\/span>\u03b1<\/span>2<\/span><\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>(<\/span>\u03b1<\/span>1<\/span><\/span>+<\/span>\u03b1<\/span>2<\/span><\/span>)<\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b11A\u2192+\u03b12A\u2192=(\u03b11+\u03b12)A\u2192<\/span><\/span><\/td>\r\n<\/tr>\r\n
The component form of a vector in two dimensions<\/td>\r\nA<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>A<\/span>x<\/span><\/span>i<\/span>\u02c6<\/span><\/span><\/span><\/span>+<\/span>A<\/span>y<\/span><\/span>j<\/span>\u02c6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A\u2192=Axi^+Ayj^<\/span><\/span><\/td>\r\n<\/tr>\r\n
Scalar components of a vector in two dimensions<\/td>\r\n{<\/span>A<\/span>x<\/span><\/span>=<\/span>x<\/span>e<\/span><\/span>\u2212<\/span>x<\/span>b<\/span><\/span><\/span><\/span><\/span>A<\/span>y<\/span><\/span>=<\/span>y<\/span>e<\/span><\/span>\u2212<\/span>y<\/span>b<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>{Ax=xe\u2212xbAy=ye\u2212yb<\/span><\/span><\/td>\r\n<\/tr>\r\n
Magnitude of a vector in a plane<\/td>\r\nA<\/span>=<\/span>A<\/span>2<\/span>x<\/span><\/span>+<\/span>A<\/span>2<\/span>y<\/span><\/span><\/span><\/span>\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u221a<\/span><\/span><\/span><\/span><\/span><\/span>A=Ax2+Ay2<\/span><\/span><\/td>\r\n<\/tr>\r\n
The direction angle of a vector in a plane<\/td>\r\n\u03b8<\/span>A<\/span><\/span>=<\/span>tan<\/span><\/span>\u22121<\/span><\/span><\/span>(<\/span>A<\/span>y<\/span><\/span><\/span>A<\/span>x<\/span><\/span><\/span><\/span><\/span>)<\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b8A=tan\u22121(AyAx)<\/span><\/span><\/td>\r\n<\/tr>\r\n
Scalar components of a vector in a plane<\/td>\r\n{<\/span>A<\/span>x<\/span><\/span>=<\/span>A<\/span><\/span>cos<\/span><\/span>\u03b8<\/span>A<\/span><\/span><\/span><\/span><\/span>A<\/span>y<\/span><\/span>=<\/span>A<\/span><\/span>sin<\/span><\/span>\u03b8<\/span>A<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>{Ax=Acos\u03b8AAy=Asin\u03b8A<\/span><\/span><\/td>\r\n<\/tr>\r\n
Polar coordinates in a plane<\/td>\r\n{<\/span>x<\/span>=<\/span>r<\/span><\/span>cos<\/span><\/span>\u03c6<\/span><\/span><\/span><\/span>y<\/span>=<\/span>r<\/span><\/span>sin<\/span><\/span>\u03c6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>{x=rcos\u03c6y=rsin\u03c6<\/span><\/span><\/td>\r\n<\/tr>\r\n
The component form of a vector in three dimensions<\/td>\r\nA<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>A<\/span>x<\/span><\/span>i<\/span>\u02c6<\/span><\/span><\/span><\/span>+<\/span>A<\/span>y<\/span><\/span>j<\/span>\u02c6<\/span><\/span><\/span><\/span>+<\/span>A<\/span>z<\/span><\/span>k<\/span>\u02c6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A\u2192=Axi^+Ayj^+Azk^<\/span><\/span><\/td>\r\n<\/tr>\r\n
The scalar\u00a0z<\/em>-component of a vector in three dimensions<\/td>\r\nA<\/span>z<\/span><\/span>=<\/span>z<\/span>e<\/span><\/span>\u2212<\/span>z<\/span>b<\/span><\/span><\/span><\/span><\/span><\/span><\/span>Az=ze\u2212zb<\/span><\/span><\/td>\r\n<\/tr>\r\n
Magnitude of a vector in three dimensions<\/td>\r\nA<\/span>=<\/span>A<\/span>2<\/span>x<\/span><\/span>+<\/span>A<\/span>2<\/span>y<\/span><\/span>+<\/span>A<\/span>2<\/span>z<\/span><\/span><\/span><\/span>\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u203e\u221a<\/span><\/span><\/span><\/span><\/span><\/span>A=Ax2+Ay2+Az2<\/span><\/span><\/td>\r\n<\/tr>\r\n
Distributive property<\/td>\r\n\u03b1<\/span>(<\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>+<\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>)<\/span>=<\/span>\u03b1<\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>+<\/span>\u03b1<\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u03b1(A\u2192+B\u2192)=\u03b1A\u2192+\u03b1B\u2192<\/span><\/span><\/td>\r\n<\/tr>\r\n
Antiparallel vector to\u00a0A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A\u2192<\/span><\/span><\/td>\r\n\u2212<\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>\u2212<\/span>A<\/span>x<\/span><\/span>i<\/span>\u02c6<\/span><\/span><\/span><\/span>\u2212<\/span>A<\/span>y<\/span><\/span>j<\/span>\u02c6<\/span><\/span><\/span><\/span>\u2212<\/span>A<\/span>z<\/span><\/span>k<\/span>\u02c6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>\u2212A\u2192=\u2212Axi^\u2212Ayj^\u2212Azk^<\/span><\/span><\/td>\r\n<\/tr>\r\n
Equal vectors<\/td>\r\nA<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span>\u21d4<\/span><\/span>\u23a7\u23a9\u23a8\u23aa\u23aa<\/span>A<\/span>x<\/span><\/span>=<\/span>B<\/span>x<\/span><\/span><\/span><\/span><\/span>A<\/span>y<\/span><\/span>=<\/span>B<\/span>y<\/span><\/span><\/span><\/span><\/span>A<\/span>z<\/span><\/span>=<\/span>B<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A\u2192=B\u2192\u21d4{Ax=BxAy=ByAz=Bz<\/span><\/span><\/td>\r\n<\/tr>\r\n
Components of the resultant of\u00a0N<\/em>\u00a0vectors<\/td>\r\n\u23a7\u23a9\u23a8\u23aa\u23aa\u23aa\u23aa\u23aa\u23aa\u23aa\u23aa<\/span>F<\/span>R<\/span>x<\/span><\/span><\/span>=<\/span>\u2211<\/span>k<\/span>=<\/span>1<\/span><\/span>N<\/span><\/span>F<\/span>k<\/span>x<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span>F<\/span>1<\/span>x<\/span><\/span><\/span>+<\/span>F<\/span>2<\/span>x<\/span><\/span><\/span>+<\/span>\u2026<\/span>+<\/span>F<\/span>N<\/span>x<\/span><\/span><\/span><\/span><\/span><\/span>F<\/span>R<\/span>y<\/span><\/span><\/span>=<\/span>\u2211<\/span>k<\/span>=<\/span>1<\/span><\/span>N<\/span><\/span>F<\/span>k<\/span>y<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span>F<\/span>1<\/span>y<\/span><\/span><\/span>+<\/span>F<\/span>2<\/span>y<\/span><\/span><\/span>+<\/span>\u2026<\/span>+<\/span>F<\/span>N<\/span>y<\/span><\/span><\/span><\/span><\/span><\/span>F<\/span>R<\/span>z<\/span><\/span><\/span>=<\/span>\u2211<\/span>k<\/span>=<\/span>1<\/span><\/span>N<\/span><\/span>F<\/span>k<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span>=<\/span>F<\/span>1<\/span>z<\/span><\/span><\/span>+<\/span>F<\/span>2<\/span>z<\/span><\/span><\/span>+<\/span>\u2026<\/span>+<\/span>F<\/span>N<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>{FRx=\u2211k=1NFkx=F1x+F2x+\u2026+FNxFRy=\u2211k=1NFky=F1y+F2y+\u2026+FNyFRz=\u2211k=1NFkz=F1z+F2z+\u2026+FNz<\/span><\/span><\/td>\r\n<\/tr>\r\n
General unit vector<\/td>\r\nV<\/span>\u02c6<\/span><\/span><\/span><\/span>=<\/span>V<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>V<\/span><\/span><\/span><\/span><\/span><\/span><\/span>V^=V\u2192V<\/span><\/span><\/td>\r\n<\/tr>\r\n
Definition of the scalar product<\/td>\r\nA<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>\u00b7<\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>A<\/span>B<\/span><\/span>cos<\/span><\/span>\u03c6<\/span><\/span><\/span><\/span><\/span><\/span>A\u2192\u00b7B\u2192=ABcos\u03c6<\/span><\/span><\/td>\r\n<\/tr>\r\n
Commutative property of the scalar product<\/td>\r\nA<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>\u00b7<\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>\u00b7<\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A\u2192\u00b7B\u2192=B\u2192\u00b7A\u2192<\/span><\/span><\/td>\r\n<\/tr>\r\n
Distributive property of the scalar product<\/td>\r\nA<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>\u00b7<\/span>(<\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>+<\/span>C<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>)<\/span>=<\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>\u00b7<\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>+<\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>\u00b7<\/span>C<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A\u2192\u00b7(B\u2192+C\u2192)=A\u2192\u00b7B\u2192+A\u2192\u00b7C\u2192<\/span><\/span><\/td>\r\n<\/tr>\r\n
Scalar product in terms of scalar components of vectors<\/td>\r\nA<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>\u00b7<\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>A<\/span>x<\/span><\/span>B<\/span>x<\/span><\/span>+<\/span>A<\/span>y<\/span><\/span>B<\/span>y<\/span><\/span>+<\/span>A<\/span>z<\/span><\/span>B<\/span>z<\/span><\/span><\/span><\/span><\/span><\/span><\/span>A\u2192\u00b7B\u2192=AxBx+AyBy+AzBz<\/span><\/span><\/td>\r\n<\/tr>\r\n
Cosine of the angle between two vectors<\/td>\r\ncos<\/span><\/span>\u03c6<\/span>=<\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>\u00b7<\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span>A<\/span>B<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>cos\u03c6=A\u2192\u00b7B\u2192AB<\/span><\/span><\/td>\r\n<\/tr>\r\n
Dot products of unit vectors<\/td>\r\ni<\/span>\u02c6<\/span><\/span><\/span><\/span>\u00b7<\/span>j<\/span>\u02c6<\/span><\/span><\/span><\/span>=<\/span>j<\/span>\u02c6<\/span><\/span><\/span><\/span>\u00b7<\/span>k<\/span>\u02c6<\/span><\/span><\/span><\/span>=<\/span>k<\/span>\u02c6<\/span><\/span><\/span><\/span>\u00b7<\/span>i<\/span>\u02c6<\/span><\/span><\/span><\/span>=<\/span>0<\/span><\/span><\/span><\/span><\/span><\/span>i^\u00b7j^=j^\u00b7k^=k^\u00b7i^=0<\/span><\/span><\/td>\r\n<\/tr>\r\n
Magnitude of the vector product (definition)<\/td>\r\n\u2223\u2223\u2223<\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>\u2223\u2223\u2223<\/span>=<\/span>A<\/span>B<\/span><\/span>sin<\/span><\/span>\u03c6<\/span><\/span><\/span><\/span><\/span><\/span>|A\u2192\u00d7B\u2192|=ABsin\u03c6<\/span><\/span><\/td>\r\n<\/tr>\r\n
Anticommutative property of the vector product<\/td>\r\nA<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>\u2212<\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A\u2192\u00d7B\u2192=\u2212B\u2192\u00d7A\u2192<\/span><\/span><\/td>\r\n<\/tr>\r\n
Distributive property of the vector product<\/td>\r\nA<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>(<\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>+<\/span>C<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>)<\/span>=<\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>+<\/span>A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>C<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A\u2192\u00d7(B\u2192+C\u2192)=A\u2192\u00d7B\u2192+A\u2192\u00d7C\u2192<\/span><\/span><\/td>\r\n<\/tr>\r\n
Cross products of unit vectors<\/td>\r\n\u23a7\u23a9\u23a8\u23aa\u23aa<\/span>i<\/span>\u02c6<\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>j<\/span>\u02c6<\/span><\/span><\/span><\/span>=<\/span>+<\/span>k<\/span>\u02c6<\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>j<\/span>\u02c6<\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>k<\/span>\u02c6<\/span><\/span><\/span><\/span>=<\/span>+<\/span>i<\/span>\u02c6<\/span><\/span><\/span><\/span>,<\/span><\/span><\/span>k<\/span>\u02c6<\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>i<\/span>\u02c6<\/span><\/span><\/span><\/span>=<\/span>+<\/span>j<\/span>\u02c6<\/span><\/span><\/span><\/span>.<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>{i^\u00d7j^=+k^,j^\u00d7k^=+i^,k^\u00d7i^=+j^.<\/span><\/span><\/td>\r\n<\/tr>\r\n
The cross product in terms of scalar\r\n
<\/div>\r\ncomponents of vectors<\/td>\r\n
A<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span><\/span>\u00d7<\/span><\/span>B<\/span>\u20d7\u00a0<\/span><\/span><\/span><\/span>=<\/span>(<\/span>A<\/span>y<\/span><\/span>B<\/span>z<\/span><\/span>\u2212<\/span>A<\/span>z<\/span><\/span>B<\/span>y<\/span><\/span>)<\/span>i<\/span>\u02c6<\/span><\/span><\/span><\/span>+<\/span>(<\/span>A<\/span>z<\/span><\/span>B<\/span>x<\/span><\/span>\u2212<\/span>A<\/span>x<\/span><\/span>B<\/span>z<\/span><\/span>)<\/span>j<\/span>\u02c6<\/span><\/span><\/span><\/span>+<\/span>(<\/span>A<\/span>x<\/span><\/span>B<\/span>y<\/span><\/span>\u2212<\/span>A<\/span>y<\/span><\/span>B<\/span>x<\/span><\/span>)<\/span>k<\/span>\u02c6<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>A\u2192\u00d7B\u2192=(AyBz\u2212AzBy)i^+(AzBx\u2212AxBz)j^+(AxBy\u2212AyBx)k^<\/span><\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/section><\/div>\r\n \r\n\r\n<\/div>\r\n
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Summary<\/span><\/h3>\r\n
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2.1<\/span>\u00a0<\/span>Scalars and Vectors<\/span><\/h4>\r\n