Umkhiqizo we-Dot usebenzisa izingxenye ze-vector zeyunithi

Izinto Zokuphindaphinda Amachashazi Ezisebenzisa Izingxenye Zevektha Yeyunithi

Singabala umkhiqizo we-scalar ngqo uma sazi izingxenye ze-x, y kanye ne-z zama-vectors u-A no -B (ama-vectors aziwayo).

Ukuze senze umkhiqizo wechashazi ngale ndlela, siqala ngokwenza umkhiqizo wechashazi wamavektha eyunithi, ngemva kwalokho siveza amavektha u-A no -B ezingxenyeni zawo, sihlukanise umkhiqizo bese sisebenzisa umkhiqizo wamavektha eyunithi.

Amavekhtha eyunithi u-i , u-j, no -k aqondene, okwenza izibalo zibe lula. Sisebenzisa i-scalar multiplication equation etholakala ngenhla ( AB = AB cos theta ), sithola:

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

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

Manje siveza amavekhtha u-A no-B ngokwezakhi zawo, sihlukanise umkhiqizo wawo bese sisebenzisa umkhiqizo wamavekhtha eyunithi.

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)

UKarena i . i = j . j = k . k = 1 kanye i . j = i . k = j . k = 0, bese kuba:

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 By _y (1) + 0 +

0 + 0 + A _z B _z (1)

_A . B = A_xB_x + A_yB_y + A_zB_z

Ngokusekelwe emiphumeleni yalesi sibalo, kungaphethwa ngokuthi umkhiqizo we-scalar noma umkhiqizo we-dot wama-vector amabili yisamba semikhiqizo yezingxenye zawo ezifanayo.

Isibonelo Sombuzo 1:

Ivektha enkulu A dan B ngokulandelana ziyi-5 kanye ne-4, njengoba kuboniswe esithombeni esingezansi. I-engeli eyakhiwe ingu-90^o. Yibale umkhiqizo wamachashazi womabili ama-vector.

Ingxoxo

Ngaphambi kokuba sibale umkhiqizo wamachashazi wamavekhtha u-A no-B, sidinga kuqala ukwazi izingxenye zevekhtha yesibili.

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

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

A _z = 0

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

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

B _z = 0

Ivekhtha A inezingxenye zevekhtha kuphela ku-x-axis kanti ivekhtha B inezingxenye zevekhtha kuphela ku-y-axis. Ingxenye ka-z ingu-zero ngoba amavekhtha u- A no -B asendizeni ye-xy.

Manje sibala umkhiqizo wamachashazi phakathi kwamavekhtha u-A no -B sisebenzisa i-dot product equation namavekhtha ezingxenye:

_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

Ake siqhathanise nendlela yokuqala

AB = AB cos theta

AB = (4)(5) cos 90

I-AB = (4) (5) (0)

AB = 0

Umphumela uyafana.

Isibonelo Sombuzo 2:

^{Ivektha enkulu} A dan B zingu-5 no-4 ngokulandelana, njengoba kuboniswe esithombeni esingezansi. Bala umkhiqizo wamachashazi womabili amavekhtha, uma i-engeli eyakhiwe ingu-30^o

Ingxoxo

Ngaphambi kokuba sibale umkhiqizo wamachashazi wamavekhtha u-A no-B, sidinga kuqala ukwazi izingxenye zevekhtha yesibili.

Ingxenye ka-z ingu-zero ngoba amavektha u-A no -B asendizeni ye-xy.

Manje sibala umkhiqizo wamachashazi phakathi kwamavekhtha u-A no -B sisebenzisa i-dot product equation namavekhtha ezingxenye:

Qhathanisa nendlela yokuqala.