Cov khoom siv dot siv cov khoom vector unit

Cov Khoom Siv Dot Multiplication Siv Unit Vector Components

Peb tuaj yeem xam cov khoom scalar ncaj qha yog tias peb paub cov x, y thiab z Cheebtsam ntawm vectors A thiab B (cov vectors paub).

Yuav ua ib qho dot product li no, peb xub ua ib qho dot product ntawm cov unit vectors, tom qab ntawd peb qhia cov vectors A thiab B hauv lawv cov khoom, rhuav tshem cov khoom thiab siv cov khoom ntawm cov unit vectors.

Cov unit vectors i , j, thiab k yog perpendicular rau ib leeg, ua rau kev xam yooj yim dua. Siv cov scalar multiplication equation uas tau los saum toj no ( AB = AB cos theta ), peb tau txais:

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

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

Tam sim no peb qhia cov vectors A thiab B raws li lawv cov khoom, rhuav tshem lawv cov khoom thiab siv cov khoom ntawm cov vectors 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)

Vim i . i = j . j = k . k = 1 thiab i . j = i . k = j . k = 0, ces:

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 B _y (1) + 0 +

0 + 0 + A _z B _z (1)

_A . B = A_xB_x + A_yB_y + A_zB_z

Raws li cov txiaj ntsig ntawm qhov kev xam no, nws tuaj yeem xaus lus tias cov khoom lag luam scalar lossis cov khoom lag luam dot ntawm ob lub vectors yog qhov sib ntxiv ntawm cov khoom lag luam ntawm lawv cov khoom zoo sib xws.

Piv txwv lus nug 1:

Vector loj A dan B feem yog 5 thiab 4, raws li pom hauv daim duab hauv qab no. Lub kaum sab xis tsim yog 90^oSuav nws cov khoom dot ob lub vectors.

Kev Sib Tham

Ua ntej peb xam cov dot product ntawm cov vectors A thiab B, peb yuav tsum paub cov khoom ntawm cov vector thib ob ua ntej.

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

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

A _z = 0

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

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

Bz = ₀

Vector A tsuas muaj cov khoom vector ntawm x-axis thiab vector B tsuas muaj cov khoom vector ntawm y-axis. Lub z yog xoom vim tias vectors A thiab B nyob hauv lub dav hlau xy.

Tam sim no peb xam cov dot product ntawm vectors A thiab B siv cov dot product equation nrog cov vectors sib xyaw:

_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

Cia peb piv rau thawj txoj kev

AB = AB cos theta

AB = (4)(5) cos 90

AB = (4) (5) (0)

AB = 0

Qhov tshwm sim yog tib yam.

Piv txwv lus nug 2:

^{Vector loj} A dan B yog 5 thiab 4 feem, raws li pom hauv daim duab hauv qab no. Xam cov khoom dot ob qho tib si vectors, yog tias lub kaum sab xis tsim yog 30^o

Kev Sib Tham

Ua ntej peb xam cov dot product ntawm cov vectors A thiab B, peb yuav tsum paub cov khoom ntawm cov vector thib ob ua ntej.

Tus z Cheebtsam yog xoom vim tias cov vectors A thiab B nyob hauv lub dav hlau xy.

Tam sim no peb xam cov dot product ntawm vectors A thiab B siv cov dot product equation nrog cov vectors sib xyaw:

Piv nrog thawj txoj kev.