Exercise
https://texercises.com/exercise/cross-product/
Question
Solution
Short
Video
\(\LaTeX\)
No explanation / solution video to this exercise has yet been created.

Visit our YouTube-Channel to see solutions to other exercises.
Don't forget to subscribe to our channel, like the videos and leave comments!
Exercise:
The em vector product of two vectors veca leftmatrixa_xa_ya_zmatrixright textrm and vecb leftmatrixb_xb_yb_zmatrixright is given by leftmatrixa_y b_z - a_z b_ya_z b_x - a_x b_za_x b_y - a_y b_xmatrixright abcliste abc Show that for veca along the x axis and vecb along the y axis vecacrossvecb is a vector along the z axis. abc Show that for any two vectors veca and vecb in the xy plane vecacrossvecb is a vector along the z axis. abc Show that for any two vectors veca and vecb vecacrossvecb is perpicular to both veca and vecb. abcliste

Solution:
abcliste abc The vectors veca and vecb can be written as veca leftmatrixamatrixright textrm and vecb leftmatrixbmatrixright It follows for the cross product vecacrossvecb leftmatrix - b - a a b - matrixright leftmatrixa bmatrixright This is obviously a vector along the z axis. abc The vectors veca and vecb can be written as veca leftmatrixa_xa_ymatrixright textrm and vecb leftmatrixb_xb_ymatrixright It follows for the cross product vecacrossvecb leftmatrixa_y - b_y b_x - a_x a_x b_y - a_y b_xmatrixright leftmatrixa_x b_y - a_y b_xmatrixright This is obviously a vector along the z axis. abc Two vectors are perpicular to each other if their dot product is equal to zero. The dot product of vecacrossvecb and veca is leftvecacrossvecbright veca leftmatrixa_y b_z - a_z b_ya_z b_x - a_x b_za_x b_y - a_y b_xmatrixright leftmatrixa_xa_ya_zmatrixright a_x a_y b_z-a_x a_z b_y+a_y a_z b_x nonumber & quad - a_x a_y b_z +a_x a_z b_y - a_y a_z b_x a_y a_z-a_y a_zb_x nonumber & quad +a_x a_z-a_x a_zb_y nonumber & quad +a_x a_y-a_x a_yb_z The verification for vecb works in the same way. abcliste
Meta Information
\(\LaTeX\)-Code
Exercise:
The em vector product of two vectors veca leftmatrixa_xa_ya_zmatrixright textrm and vecb leftmatrixb_xb_yb_zmatrixright is given by leftmatrixa_y b_z - a_z b_ya_z b_x - a_x b_za_x b_y - a_y b_xmatrixright abcliste abc Show that for veca along the x axis and vecb along the y axis vecacrossvecb is a vector along the z axis. abc Show that for any two vectors veca and vecb in the xy plane vecacrossvecb is a vector along the z axis. abc Show that for any two vectors veca and vecb vecacrossvecb is perpicular to both veca and vecb. abcliste

Solution:
abcliste abc The vectors veca and vecb can be written as veca leftmatrixamatrixright textrm and vecb leftmatrixbmatrixright It follows for the cross product vecacrossvecb leftmatrix - b - a a b - matrixright leftmatrixa bmatrixright This is obviously a vector along the z axis. abc The vectors veca and vecb can be written as veca leftmatrixa_xa_ymatrixright textrm and vecb leftmatrixb_xb_ymatrixright It follows for the cross product vecacrossvecb leftmatrixa_y - b_y b_x - a_x a_x b_y - a_y b_xmatrixright leftmatrixa_x b_y - a_y b_xmatrixright This is obviously a vector along the z axis. abc Two vectors are perpicular to each other if their dot product is equal to zero. The dot product of vecacrossvecb and veca is leftvecacrossvecbright veca leftmatrixa_y b_z - a_z b_ya_z b_x - a_x b_za_x b_y - a_y b_xmatrixright leftmatrixa_xa_ya_zmatrixright a_x a_y b_z-a_x a_z b_y+a_y a_z b_x nonumber & quad - a_x a_y b_z +a_x a_z b_y - a_y a_z b_x a_y a_z-a_y a_zb_x nonumber & quad +a_x a_z-a_x a_zb_y nonumber & quad +a_x a_y-a_x a_yb_z The verification for vecb works in the same way. abcliste
Contained in these collections:
  1. 3 | 3
  2. 8 | 11

Attributes & Decorations
Branches
Magnetism
Tags
biot-savart, cross product, vector geometry
Content image
Difficulty
(2, default)
Points
0 (default)
Language
ENG (English)
Type
Calculative / Quantity
Creator by
Decoration
File
Link