Vector Algebra Essentials. Cartesian coordinate frame. Distance between point and coordinate frame origin: Direction cosines

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General Physics Institute, Russian Academy of Science
https://sites.google.com/site/drolegbshiryaev/
8-916-205-55-82 [email protected]

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RNRMU Physics/Math
Math Prelude 1: Vector Algebra Essentials
Cartesian coordinate frame
Distance between point and coordinate
frame origin:
Direction cosines:

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RNRMU Physics/Math
Math Prelude 1: Vector Algebra Essentials
Cartesian coordinate frame
Distance between point and coordinate
frame origin:
Direction cosines:

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RNRMU Physics/Math
Vector Magnitudes and Collinearity
=
Vector length:
Unit vector in same direction:
Equal vectors
Collinear vectors
Unit Vectors in Coordinate Directions
,

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RNRMU Physics/Math
Operations on Vectors in Coordinate Representation

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RNRMU Physics/Math
Dot Product
Dot Product in Terms of Vector Components

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RNRMU Physics/Math
Vector Magnitude
Vectors Orgthogonality Condition
Projection of a Vector on Another Vector's Direction
Angle Between Vectors

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RNRMU Physics/Math
Vectors on a Plane

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RNRMU Physics/Math
Examples
Example: prove that the diagonals are perpendicular for
Example: find the length of
for
Example: calculate work performed by the force
to move a body from
to
. Calculate
the angle between the force and the displacement

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RNRMU Physics/Math
Cross Product

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RNRMU Physics/Math
Cross Product
Area of triangle with sides a and b:

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RNRMU Physics/Math
Cross Product in Coordinate Terms

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RNRMU Physics/Math
Cross Product in Coordinate Terms

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RNRMU Physics/Math
Box Product
for vectors lying in the same plane

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RNRMU Physics/Math
Triple Product in Coordinate Terms
Example: find prism volume for