Cross Product
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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
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