Chogulitsa cha dot pogwiritsa ntchito zigawo za vector ya unit

Zinthu Zochulukitsa Madontho Pogwiritsa Ntchito Zigawo za Vekitala ya Unit

Tikhoza kuwerengera mwachindunji chinthu cha scalar ngati tikudziwa zigawo za x, y ndi z za ma vector A ndi B (ma vector odziwika).

Kuti tichite chinthu cha dot motere, choyamba timachita chinthu cha dot cha ma vector a unit, kenako timawonetsa ma vector A ndi B m'zigawo zawo, timawononga chinthucho ndikugwiritsa ntchito chinthu cha ma vector a unit.

Ma vector a unit i , j, ndi k ndi olunjika kwa wina ndi mnzake, zomwe zimapangitsa kuti kuwerengera kukhale kosavuta. Pogwiritsa ntchito equation ya scalar multiplication yomwe yachokera pamwambapa ( AB = AB cos theta ), timapeza:

i. i = j. j = k. k = (1)(1) cos 0 = 1

i. j = i. k = j. k = (1)(1) cos 90 ^o = 0

Tsopano tikuwonetsa ma vector A ndi B malinga ndi zigawo zawo, kuwononga zinthu zawo ndikugwiritsa ntchito zomwe zapangidwa ndi ma vector a unit.

A . B = A_xi . B_xi + A_xi . B_yj + A_xi . B_zk +

A_yj . B_xi + A_yj . B_yj + A_yj . B_zk +

A_zk . B_xi + A_zk . B_yj + A_zk . B_zk

A . B = A_xB_x (i . i) + A_xB_y (i . j) + A_x B_z (i . k) +

A_yB_x (j . i) + A_yB_y (j . j) + A_yB_z (j . k) +

A_zB_x (k . i) + A_zB_y (k . j) + A_zB_z (k . k)

Chifukwa i . i = j . j = k . k = 1 ndi i . j = i . k = j . k = 0, kenako:

A . B = A_xB_x (1) + A_xB_y (0) + A_x B_z (0) +

A_yB_x (0) + A_yB_y (1) + A_yB_z (0) +

A_zB_x (0) + A_zB_y (0) + A_zB_z (1)

A. B = A _x B _x ( 1) + 0 + 0 +

0 + A _y Pofika _y (1) + 0 +

0 + 0 + A _z B _z (1)

_A . B = A_xB_x + A_yB_y + A_zB_z

Kutengera zotsatira za kuwerengera kumeneku, zitha kutsimikiziridwa kuti chinthu cha scalar kapena chinthu cha dot cha ma vector awiri ndi chiwerengero cha zinthu zomwe zili m'zigawo zawo zofanana.

Chitsanzo cha Funso 1:

Vekitala yayikulu A dani B motsatana ndi 5 ndi 4, monga momwe chithunzi chili pansipa chikusonyezera. Ngodya yopangidwa ndi 90^o. Werengani chinthu cha dontho mavekitala onse awiri.

Zokambirana

Tisanawerengere zotsatira za ma vekitala A ndi B, choyamba tiyenera kudziwa zigawo za vekitala yachiwiri.

A _x = (5) cos 0 ^o = (5) (1) = 5

A _y = (5) tchimo 0 ^o = (5) (0) = 0

A _z = 0

B _x = (4) cos 90 ^o = (4) (0) = 0

B _y = (4) tchimo 90 ^o = (4) (1) = 4

B _z = 0

Vekitala A ili ndi zigawo za vekitala pa x-axis ndipo vekitala B ili ndi zigawo za vekitala pa y-axis. Z component ndi zero chifukwa mavekitala A ndi B ali mu xy plane.

Tsopano tikuwerengera chinthu cha dot pakati pa mavekitala A ndi B pogwiritsa ntchito equation ya chinthu cha dot ndi mavekitala a component:

_A. B = A_x B_x + A_yB_y + A_zB_z

A. B = (5) (0) + (0) (4) + 0

A. B = 0 + 0 + 0

A. B = 0

Tiyeni tiyerekezere ndi njira yoyamba

AB = AB cos theta

AB = (4)(5) cos 90

AB = (4) (5) (0)

AB = 0

Zotsatira zake ndi zomwezo.

Chitsanzo cha Funso 2:

^{Vekitala yayikulu} A dani B ndi 5 ndi 4 motsatana, monga momwe chithunzi chili pansipa chikusonyezera. chinthu cha dontho ma vector onse awiri, ngati ngodya yopangidwa ndi 30^o

Zokambirana

Tisanawerengere zotsatira za ma vekitala A ndi B, choyamba tiyenera kudziwa zigawo za vekitala yachiwiri.

Gawo la z ndi zero chifukwa ma vector A ndi B ali mu xy plane.

Tsopano tikuwerengera chinthu cha dot pakati pa mavekitala A ndi B pogwiritsa ntchito equation ya chinthu cha dot ndi mavekitala a component:

Yerekezerani ndi njira yoyamba.