Operations with Vector Functions
The following operations give rise to new vector-functions using component-wise operations on vectors.
Definition (Operations with Vector-Valued Functions) Let
![operations with vector functions _gr_1.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_1.gif){kind=small}
and
![operations with vector functions _gr_2.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_2.gif){kind=small}
be vector-valued functions of the real variable
![operations with vector functions _gr_3.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_3.gif){kind=small}
, and let
![operations with vector functions _gr_4.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_4.gif){kind=small}
be a real-valued function. Then
![operations with vector functions _gr_5.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_5.gif){kind=small}
![operations with vector functions _gr_6.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_6.gif){kind=small}
![operations with vector functions _gr_7.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_7.gif){kind=small}
![operations with vector functions _gr_8.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_8.gif){kind=small}
and
![operations with vector functions _gr_9.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_9.gif){kind=small}
are defined as follows:
![operations with vector functions _gr_10.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_10.gif){kind=small}
![operations with vector functions _gr_11.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_11.gif){kind=small}
![operations with vector functions _gr_12.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_12.gif){kind=small}
These operations are defined on the intersection of the domain of the vector-valued and real-valued functions that occur in the definitions, respectively.
Example (Operations with Vector-Valued Functions) Given
![operations with vector functions _gr_13.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_13.gif){kind=small}
and
![operations with vector functions _gr_14.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_14.gif){kind=small}
find the vector function
![operations with vector functions _gr_15.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_15.gif){kind=small}
Solution. We have
![operations with vector functions _gr_16.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_16.gif)
![operations with vector functions _gr_17.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_17.gif){kind=small}
![operations with vector functions _gr_18.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_18.gif){kind=small}
![operations with vector functions _gr_19.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_19.gif){kind=small}
Example (Operations with Vector-Valued Functions) Given the vector-valued functions
![operations with vector functions _gr_20.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_20.gif){kind=small}
![operations with vector functions _gr_21.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_21.gif){kind=small}
![operations with vector functions _gr_22.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_22.gif){kind=small}
find
![operations with vector functions _gr_23.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_23.gif){kind=small}
Solution. We have,
![operations with vector functions _gr_24.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_24.gif){kind=small}
![operations with vector functions _gr_25.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_25.gif){kind=small}
![operations with vector functions _gr_26.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_26.gif){kind=small}
![operations with vector functions _gr_27.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_27.gif){kind=small}
Example (Operations with Vector-Valued Functions) Given the vector-valued functions
![operations with vector functions _gr_28.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_28.gif){kind=small}
![operations with vector functions _gr_29.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_29.gif){kind=small}
![operations with vector functions _gr_30.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_30.gif){kind=small}
find
![operations with vector functions _gr_31.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_31.gif){kind=small}
Solution. We have,
![operations with vector functions _gr_32.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_32.gif){kind=small}
![operations with vector functions _gr_33.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_33.gif)
![operations with vector functions _gr_34.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_34.gif){kind=small}
![operations with vector functions _gr_35.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_35.gif){kind=small}
![operations with vector functions _gr_36.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_36.gif){kind=small}
![operations with vector functions _gr_37.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_37.gif){kind=small}
Recommended Links
![operations with vector functions _gr_38.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_38.gif){kind=small}
vector functions
![operations with vector functions _gr_39.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_39.gif){kind=small}
operations with vector functions
![operations with vector functions _gr_40.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_40.gif){kind=small}
limits and continuity of vector functions
![operations with vector functions _gr_41.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_41.gif){kind=small}
graphs of vector functions
![operations with vector functions _gr_42.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_42.gif){kind=small}
vector differentiation
![operations with vector functions _gr_43.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_43.gif){kind=small}
tangent vectors
![operations with vector functions _gr_44.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_44.gif){kind=small}
smooth curves
![operations with vector functions _gr_45.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_45.gif){kind=small}
derivative rules
![operations with vector functions _gr_46.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_46.gif){kind=small}
vector integration
![operations with vector functions _gr_47.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_47.gif){kind=small}
motion of an object
![operations with vector functions _gr_48.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_48.gif){kind=small}
unit tangent and unit normal vectors
![operations with vector functions _gr_49.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_49.gif){kind=small}
arc length function
![operations with vector functions _gr_50.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_50.gif){kind=small}
curvature
![operations with vector functions _gr_51.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_51.gif){kind=small}
Recommended Math Books
![operations with vector functions _gr_52.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_52.gif){kind=small}
Thomas' Calculus, Early Transcendentals, Media Upgrade (11th Edition)
![operations with vector functions _gr_53.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_53.gif){kind=small}
Thomas' Calculus, Media Upgrade (11th Edition)
![operations with vector functions _gr_54.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_54.gif){kind=small}
Thomas' Calculus Early Transcendentals; Student's Solutions Manual; Part One
![operations with vector functions _gr_55.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_55.gif){kind=small}
Calculus (With Analytic Geometry)(8th edition)
![operations with vector functions _gr_56.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_56.gif){kind=small}
Calculus (Stewart's Calculus Series)
![operations with vector functions _gr_57.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_57.gif){kind=small}
Applied Calculus
![operations with vector functions _gr_58.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_58.gif){kind=small}
Calculus Textbooks
![operations with vector functions _gr_59.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_59.gif){kind=small}
Elementary Calculus
![operations with vector functions _gr_60.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_60.gif){kind=small}
Advanced Calculus
![operations with vector functions _gr_61.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_61.gif){kind=small}
Supplementary Resources
![operations with vector functions _gr_62.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_62.gif){kind=small}
Recommended Math Gifts
![operations with vector functions _gr_63.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_63.gif){kind=small}
Math Happy
![operations with vector functions _gr_64.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_64.gif){kind=small}
Calculus Happy
![operations with vector functions _gr_65.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_65.gif){kind=small}
Homework Happy
![operations with vector functions _gr_66.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_66.gif){kind=small}
Limits Happy
![operations with vector functions _gr_67.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_67.gif){kind=small}
I Love Math
![operations with vector functions _gr_68.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_68.gif){kind=small}
I Love Calculus
![operations with vector functions _gr_69.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_69.gif){kind=small}
I Love Homework
![operations with vector functions _gr_70.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_70.gif){kind=small}
I Love Multivariate Calculus
![operations with vector functions _gr_71.gif]](pages/operations-with-vector-functions/Images/operations-with-vector-functions_gr_71.gif){kind=small}
I Love Limits
Operations With Vector Functions
Published by Library of Math -- Online math organized by subject into topics.
Written by Smith, David A.
http://www.libraryofmath.com/operations-with-vector-functions.html
