Foromo ea Vector e Resultant: Mathata a Khopolo-taba, Mokhoa le Mohlala
Vekthara ke bongata bo nang le boholo le tataiso. Fisiks le lipalo, li-vekthara hangata li sebelisoa ho hlalosa liketsahalo tse fapaneng tse kang lebelo, matla le ho falla. Ho bala vekthara e hlahang, kakaretso ea li-vekthara tse peli kapa ho feta, ke tsebo ea bohlokoa e sebelisoang khafetsa lits'ebetsong tse fapaneng tsa saense le tsa botekgeniki. Sengoloa sena se tla tšohla mohopolo oa motheo oa li-vekthara, mekhoa ea ho bala vekthara e hlahang, le ho fana ka mehlala e 'maloa ea mathata ho hlakisa kutloisiso.
Ho Utloisisa Li-vector le Li-vector tse Hlahang
Vektor
Vekthara ke ntho ea lipalo e nang le litšobotsi tse peli tse ka sehloohong:
1. Boholo: Boholo ba boleng ba vekthara.
2. Tsela: Tsela eo vekthara e shebanang le yona e bontsha tsela eo vekthara e shebaneng le yona sebakeng.
Hangata li-vector li hlalosoa e le metsu, moo bolelele ba motsu bo emelang boholo 'me tataiso ea motsu e bontša tataiso ea vector.
Vekthara e Hlahisang
Vektha e hlahang ke vektha e le 'ngoe e emelang motsoako oa livektha tse peli kapa ho feta. Mokhoa oa ho eketsa livektha o boetse o tsejoa e le "ho eketsa vektha." Ho na le mekhoa e 'maloa e ka sebelisoang ho bala vektha e hlahang, ho kenyeletsoa mekhoa ea litšoantšo le ea tlhahlobo.
Mokhoa oa ho Bala Liphetho tsa Vektara
Mokhoa oa Litšoantšo
Mokhoa oa litšoantšo o kenyelletsa ho emela li-vector ka sebopeho sa jeometri le ho sebelisa melao ea ho eketsa vector ho fumana sephetho. Melao e 'meli e meholo ea mokhoa oa litšoantšo ke:
1. Mokhoa oa Khutlotharo: Mokhoeng ona, vekthara ea bobeli e huloa ho tloha pheletsong ea vekthara ea pele. Vekthara e hlahang ke vekthara e huloang ho tloha moo vekthara ea pele e qalang ho ea qetellong ea vekthara ea bobeli.
2. Mokhoa oa Poligone: Mokhoa ona o sebelisetsoa ho eketsa li-vector tse fetang tse peli. Li-vector li huloa ka tatellano ho tloha pheletsong e 'ngoe ho ea ho e 'ngoe, 'me vector e hlahang ke vector e hokahanyang ntlha ea pele ea vector ho ea qetellong ea vector ea ho qetela.
Mokhoa oa Tlhahlobo
Mokhoa oa tlhahlobo o kenyelletsa tšebeliso ea lipalo le trigonometry ho bala vector e hlahang. Mekhoa e 'meli e meholo mokhoeng oa tlhahlobo ke:
1. Mokhoa oa Likarolo: Mokhoeng ona, vekthara e 'ngoe le e 'ngoe e arotsoe ka likarolo tsa eona ho latela li-x- le li-y-axes. Likarolo tsena li kopanngoa hammoho ho fumana likarolo tsa vekthara e hlahang. Qetellong, vekthara e hlahang e baloa ho sebelisoa theorem ea Pythagorean le trigonometry.
2. Mokhoa oa Cosine: Mokhoa ona o sebelisoa ha boholo ba li-vector tse peli le sekhutlo se pakeng tsa tsona li tsejoa. Foromo ea cosine e sebelisoa ho bala boholo ba vector e hlahang.
Liforomo tsa Vector tse Hlahisang
Mokhoa oa Likarolo
Bakeng sa divekthara tse pedi \(\mathbf{A}\) le \(\mathbf{B}\) tse nang le dikarolo:
\[
\mathbf{A} = A_x \hat{i} + A_y \hat{j}
\]
\[
\mathbf{B} = B_x \hat{i} + B_y \hat{j}
\]
Vekthara e hlahang \(\mathbf{R}\) ke:
\[
\mathbf{R} = \mathbf{A} + \mathbf{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j}
\]
Boholo ba vekthara e hlahang \(\mathbf{R}\) bo ka balwa ho sebediswa theorem ya Pythagorean:
\[
|\mathbf{R}| = \sqrt{(A_x + B_x)^2 + (A_y + B_y)^2}
\]
Tsela eo vekthara e hlahang e lebisoa ka yona e thehwa ke angle \(\theta\) e bopilweng ka x-axis:
\[
\theta = \tan^{-1}\left(\frac{A_y + B_y}{A_x + B_x}\right)
\]
Mokhoa oa Cosine
Haeba divekthara tse pedi \(\mathbf{A}\) le \(\mathbf{B}\) di na le boholo \(A\) le \(B\) le sekhutlo \(\theta\) pakeng tsa tsona, boholo ba vekthara e hlahang \(\mathbf{R}\) ke:
\[
|\mathbf{R}| = \sqrt{A^2 + B^2 + 2AB \cos \theta}
\]
Tsela eo vekthara e hlahang ka yona e ka balwa ho sebediswa foromo ya trigonometric:
\[
\tan \alpha = \frac{B \sin \theta}{A + B \cos \theta}
\]
Moo \(\alpha\) e leng sekhutlo se bopilweng ke vektha e hlahang ka vektha \(\mathbf{A}\).
Mohlala oa Bothata ba Vektara bo Hlahang
Mohlala oa Potso ea 1: Mokhoa oa Likarolo
Potso:
Livekthara tse peli \(\mathbf{A}\) le \(\mathbf{B}\) li na le likarolo tse latelang:
\[
\mathbf{A} = 3\hat{i} + 4\hat{j}
\]
\[
\mathbf{B} = 1\hat{i} + 2\hat{j}
\]
Bala vekthara e hlahang \(\mathbf{R}\).
Tharollo:
1. Kenya likarolo tse ho li-axe tsa x le y:
\[
R_x = A_x + B_x = 3 + 1 = 4
\]
\[
R_y = A_y + B_y = 4 + 2 = 6
\]
2. Bala boholo ba vekthara e hlahang:
\[
|\mathbf{R}| = \sqrt{R_x^2 + R_y^2} = \sqrt{4^2 + 6^2} = \sqrt{16 + 36} = \sqrt{52} = 7,21
\]
3. Bala tataiso ea vekthara e hlahang:
\[
\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{6}{4}\right) = \tan^{-1}(1,5) = 56,31^\circ
\]
Kahoo, vekthara e hlahang \(\mathbf{R}\) e na le boholo ba 7,21 le tataiso ea likhato tse 56,31 ho ea ho x-axis.
Mohlala oa Potso ea 2: Mokhoa oa Cosine
Potso:
Livekthara tse peli \(\mathbf{A}\) le \(\mathbf{B}\) li na le liyuniti tsa boholo \(A = 5\), \(B = 7\) liyuniti, 'me sekhutlo se pakeng tsa tsona ke 60°. Bala boholo ba vekthara e hlahang \(\mathbf{R}\).
Tharollo:
1. Sebelisa foromo ea cosine ho bala boholo ba vekthara e hlahang:
\[
|\mathbf{R}| = \sqrt{A^2 + B^2 + 2AB \cos \theta}
\]
\[
|\mathbf{R}| = \sqrt{5^2 + 7^2 + 2 \cdot 5 \cdot 7 \cdot \cos 60^\circ}
\]
\[
|\mathbf{R}| = \sqrt{25 + 49 + 70 \cdot 0,5}
\]
\[
|\mathbf{R}| = \ sqrt{25 + 49 + 35}
\]
\[
|\mathbf{R}| = \sqrt{109} = 10,44 \, \mongolo{yuniti}
\]
Kahoo, boholo ba vekthara e hlahang \(\mathbf{R}\) ke diyuniti tse 10,44.
Mohlala oa 3: Liphetho tsa Li-vector tse Tharo
Potso:
Livekthara tse tharo \(\mathbf{A}\), \(\mathbf{B}\), le \(\mathbf{C}\) li na le likarolo tse latelang:
\[
\mathbf{A} = 2\hat{i} + 3\hat{j}
\]
\[
\mathbf{B} = -1\hat{i} + 4\hat{j}
\]
\[
\mathbf{C} = 3\hat{i} – 2\hat{j}
\]
Bala vekthara e hlahang \(\mathbf{R}\).
Tharollo:
1. Kenya likarolo tse ho li-axe tsa x le y:
\[
R_x = A_x + B_x + C_x = 2 – 1 + 3 = 4
\]
\[
R_y = A_y + B_y + C_y = 3 + 4 – 2 = 5
\]
2. Bala boholo ba vekthara e hlahang:
\[
|\mathbf{R}| = \sqrt{R_x^2 + R_y^2} = \sqrt{4^2 + 5^2} = \sqrt{16 + 25} = \sqrt{41} = 6,4
\]
3. Bala tataiso ea vekthara e hlahang:
\[
\theta = \tan^{-1}\left(\frac{R_y}{R_x}\right) = \tan^{-1}\left(\frac{5}{4}\right) = \tan^{-1}(1,25) = 51,34^\circ
\]
Kahoo, vekthara e hlahang \(\mathbf{
R}\) e na le boholo ba 6,4 le tataiso ea likhato tse 51,34 ho ea ho x-axis.
Qetello
Ho bala sephetho sa vektara ke bokgoni ba bohlokwa fisiks le dipalo. Ka ho sebedisa mekgwa ya ditshwantsho kapa ya tlhahlobo, re ka fumana sephetho sa divektara tse pedi kapa ho feta. Mokgwa wa dikarolo le mokgwa wa di-cosine ke mekgwa e mmedi ya bohlokwa dipalong tsa tlhahlobo e re dumellang ho bala ka nepo boholo le tataiso ya vektara e hlahang. Mehlala e ka hodimo e bontsha tshebediso e sebetsang ya mehopolo ena, e re thusang ho utlwisisa le ho sebedisa divektara maemong a fapaneng a saense le a botekgeniki.
