Lipotso tsa Mehlala le Puisano ea Likarolo tsa Vector
Li-vector ke mohopolo oa motheo fisiks le lipalo, hangata li sebelisoa ho hlalosa bongata ka boholo le tataiso. Kutloisiso e felletseng ea li-vector e bohlokoa bakeng sa ho rarolla mathata a fapaneng saenseng le boenjiniere. Sehlooho sena se tla tšohla mehlala e 'maloa ea mathata a amanang le likarolo tsa vector, hammoho le litlhaloso tsa tsona.
Selelekela ho Li-vector
Vekthara ke bongata bo nang le litšobotsi tse peli tse ka sehloohong: boholo le tataiso. Mohlala, lebelo ke bongata ba vekthara hobane e na le boholo (ka lebelo le lekae) le tataiso (moo e eang teng). Ho emela livekthara, hangata re sebelisa metsu, moo bolelele ba motsu bo emelang boholo ba oona le tataiso ea motsu e bontšang tataiso ea oona.
Vekthara sebakeng sa mahlakore a mabedi hangata e hlaloswa e le 𝐀 = 𝑎ᵢ + 𝑏ⱼ, moo 𝑎 le 𝑏 e leng dikarolo tsa vekthara ho latela di-x- le di-y-axes, mme 𝐢 le 𝐣 ke di-vekthara tsa yuniti ho latela di-x- le di-y-axes.
Mohlala oa Potso ea 1: Ho Fumana Likarolo tsa Vekthara ho tsoa ho Setšoantšo sa Litšoantšo
Potso: Vekthara 𝐀 e na le ntlha ea qalo qalong (0,0) le ntlha ea qetello ho likhokahano (4,3). Fumana likarolo tsa vekthara 𝐀.
Puisano: Vekthara e qalang ho tloha ntlheng ea qalo (0,0) ho isa ntlheng ea qetello (4,3) e ka ngoloa ka sebopeho sa karolo e le 𝐀 = 4𝐢 + 3𝐣. Karolo e pela x-axis ke 4 'me e pela y-axis ke 3.
Mohlala oa Potso ea 2: Ho Fumana Boholo ba Vektara
Bothata: Bala boholo ba vekthara 𝐀 = 4𝐢 + 3𝐣.
Puisano: Boholo (kapa boholo) ba vekthara 𝐀 bo ka balwa ho sebediswa foromo ya Pythagorean, e leng:
\[ |𝐀| = \sqrt{𝑎² + 𝑏²} \]
Bakeng sa vekthara 𝐀 = 4𝐢 + 3𝐣, joale:
\[ |𝐀| = \sqrt{4² + 3²} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
Kahoo, boholo ba vekthara 𝐀 ke diyuniti tse 5.
Mohlala oa 3: Ho eketsa livektara tse peli
Potso: Ho fanoe ka livekthara tse peli 𝐁 = 2𝐢 + 3𝐣 le 𝐂 = -𝐢 + 4𝐣. Fumana kakaretso ea livekthara 𝐁 le 𝐂.
Puisano: Ho eketsa divekthara tse pedi, re eketsa feela dikarolo ka lehlakoreng le le leng la vekthara ka nngwe:
\[ 𝐁 + 𝐂 = (2𝐢 + 3𝐣) + (-𝐢 + 4𝐣) \]
\[ = (2 + (-1))𝐢 + (3 + 4)𝐣 \]
\[ = 1 + 7
Kahoo, sephetho sa ho eketsa divekthara 𝐁 le 𝐂 ke 𝐃 = 𝐢 + 7𝐣.
Mohlala Potso ea 4: Ho Bala Sekhutlo Pakeng tsa Li-vector tse peli
Bothata: Ha ho fanoe ka livekthara tse peli 𝐀 = 3𝐢 + 4𝐣 le 𝐁 = 4𝐢 – 3𝐣. Bala sekhutlo se pakeng tsa livekthara tse peli.
Puisano: Sekhutlo se pakeng tsa divekthara tse pedi se ka balwa ho sebediswa foromo ya cosine:
\[ \cos(𝜃) = \frac{𝐀 · 𝐁}{|𝐀| |𝐁|} \]
1. Bala sehlahisoa sa matheba (𝐀 · 𝐁):
\[ 𝐀 · 𝐁 = (3𝐢 + 4𝐣) · (4𝐢 – 3𝐣) \]
\[ = (3 4) + (4 -3) \]
\[ = 12 – 12 \]
\[ = 0 \]
2. Bala boholo ba li-vector 𝐀 le 𝐁:
\[ |𝐀| = \sqrt{3² + 4²} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
\[ |𝐁| = \sqrt{4² + (-3)²} = \sqrt{16 + 9} = \sqrt{25} = 5 \]
3. Kenya sebakeng sa foromo ea cosine:
\[ \cos(𝜃) = \frac{0}{5 5} = 0 \]
Kaha \(\cos(𝜃) = 0\), joale \(𝜃 = 90°\). Kahoo, sekhutlo se pakeng tsa livekthara tse peli ke likhato tse 90.
Mohlala oa Potso ea 5: Ho Bala Sehlahisoa se Tšepahalang sa Li-Vector
Bothata: Ha o fuwa divekthara tse pedi ka ditekanyo tse tharo, 𝐀 = 𝐢 + 2𝐣 + 3𝐤 le 𝐁 = 4𝐢 + 5𝐣 + 6𝐤, bala vekthara ya sehlahiswa se kopaneng 𝐀 × 𝐁.
Puisano: Sehlahisoa se kopaneng sa livekthara tse peli ka litekanyo tse tharo (𝐀 × 𝐁) ke:
\[ 𝐀 × 𝐁 = \begin{vmatrix} 𝐢 & 𝐣 & 𝐤 \\ 1 & 2 & 3 \\ 4 & 5 & 6 \end{vmatrix} \]
\[ = 𝐢 (2 6 – 3 5) – 𝐣 (1 6 – 3 4) + 𝐤 (1 5 – 2 4) \]
\[ = 𝐢 (12 – 15) – 𝐣 (6 – 12) + 𝐤 (5 – 8) \]
\[ = 𝐢 (-3) – 𝐣 (-6) + 𝐤 (-3) \]
\[ = -3𝐢 + 6𝐣 – 3𝐤 \]
Kahoo, sephetho sa sehlahisoa se kopaneng 𝐀 × 𝐁 ke -3𝐢 + 6𝐣 – 3𝐤.
Qetello
Fisiks le lipalo, li-vector ke mokhoa o thusang haholo oa ho emela bongata bo nang le tataiso le boholo. Ka ho utloisisa mokhoa oa ho fumana likarolo tsa li-vector, ho bala boholo, ho eketsa li-vector, le ho bala likhutlo lipakeng tsa li-vector le lihlahisoa tse tšekaletseng, re ka rarolla mathata a fapaneng a amang li-vector. Puisano ea mathata a mohlala a kaholimo e ikemiselitse ho thusa ho tebisa kutloisiso ea rona ea mohopolo ona. Qetellong, bokhoni ba ho utloisisa le ho sebetsa le li-vector ke bokhoni bo thusang haholo mafapheng a fapaneng a saense le boenjiniere.
