Mienzaniso yemibvunzo inokurukura nezveZvikamu zveVector

Mibvunzo yeMienzaniso neKukurukurirana kweZvikamu zveVector

Mavector ipfungwa huru mufizikisi nemasvomhu, anowanzo shandiswa kutsanangura huwandu nehukuru negwara. Kunzwisisa kwakakwana mavector kwakakosha pakugadzirisa matambudziko akasiyana-siyana musainzi neinjiniya. Chinyorwa chino chichakurukura mienzaniso yakati wandei yematambudziko ane chekuita nezvikamu zvevector, pamwe chete netsananguro dzazvo.

Nhanganyaya kuVectors

Vector huwandu hune hunhu huviri hukuru: hukuru negwara. Semuenzaniso, kumhanya kwemhepo huwandu hwevector nekuti ine hukuru (kukurumidza) uye gwara (kwairi kuenda). Kuti timiririre vectors, tinowanzo shandisa miseve, apo kureba kwemuseve kunomiririra hukuru hwayo uye gwara remuseve rinoratidza gwara rayo.

Vector iri munzvimbo ine mativi maviri inowanzo ratidzwa se 𝐀 = 𝑎ᵢ + 𝑏ⱼ, apo 𝑎 na 𝑏 zviri zvikamu zvevector iri padivi pe x- na y-axes, uye 𝐢 na 𝐣 zviri mayuniti vector ari padivi pe x- na y-axes.

Muenzaniso Mubvunzo 1: Kusarudza Zvikamu zveVector kubva kuMufananidzo weMifananidzo

Mubvunzo: Vekitari 𝐀 ine pokutangira pakutanga (0,0) uye pokugumira pazvikochekedzo (4,3). Sarudza zvikamu zvevekitari 𝐀.

Kukurukurirana: Vekitari inotangira kubva panotangira (0,0) kusvika panogumira (4,3) inogona kunyorwa muchimiro chechikamu se 𝐀 = 4𝐢 + 3𝐣. Chikamu chiri padivi pe x-axis i4 uye chiri padivi pe y-axis i3.

Muenzaniso Mubvunzo 2: Kuziva Hukuru hweVector

Dambudziko: Verenga hukuru hwevhekita 𝐀 = 4𝐢 + 3𝐣.

Kukurukurirana: Hukuru (kana saizi) hwevector 𝐀 hunogona kuverengerwa uchishandisa fomura yePythagorean, inoti:

\[ |𝐀| = \sqrt{𝑎² + 𝑏²} \]

Pa vhekitari 𝐀 = 4𝐢 + 3𝐣, zvino:

\[ |𝐀| = \sqrt{4² + 3²} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

Saka, hukuru hwevector 𝐀 mayunitsi mashanu.

Muenzaniso 3: Kuwedzera Mavector maviri

Mubvunzo: Kana wapiwa mavector maviri 𝐁 = 2𝐢 + 3𝐣 uye 𝐂 = -𝐢 + 4𝐣. Sarudza huwandu hwemavector 𝐁 uye 𝐂.

Kukurukurirana: Kuti tiwedzere mavekitari maviri, tinongowedzera zvikamu zvacho nenzira imwe chete yevekitari yega yega:

\[ 𝐁 + 𝐂 = (2𝐢 + 3𝐣) + (-𝐢 + 4𝐣) \]

\[ = (2 + (-1)) 𝐢 + (3 + 4) 𝐣 \]

\[ = 1 + 7 \]

Saka, mhedzisiro yekuwedzera mavector 𝐁 uye 𝐂 ndeye 𝐃 = 𝐢 + 7𝐣.

Muenzaniso Mubvunzo 4: Kuverenga Angle Pakati Pemavector Maviri

Dambudziko: Kana wapiwa mavector maviri 𝐀 = 3𝐢 + 4𝐣 uye 𝐁 = 4𝐢 – 3𝐣. Verenga kona iri pakati pemavector maviri.

Kukurukurirana: Kona iri pakati pemaveki maviri inogona kuverengerwa uchishandisa fomura yecosine:

\[ \cos(𝜃) = \frac{𝐀 · 𝐁}{|𝐀| |𝐁|} \]

1. Verenga chigadzirwa chine madotsi (𝐀 · 𝐁):

\[ 𝐀 · 𝐁 = (3𝐢 + 4𝐣) · (4𝐢 – 3𝐣) \]

\[ = (3 4) + (4 -3) \]

\[ = 12 – 12 \]

\[ = 0 \]

2. Verengai hukuru hwemavector 𝐀 uye 𝐁:

\[ |𝐀| = \sqrt{3² + 4²} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

\[ |𝐁| = \sqrt{4² + (-3)²} = \sqrt{16 + 9} = \sqrt{25} = 5 \]

3. Isa mufomura yecosine:

\[ \cos(𝜃) = \frac{0}{5 5} = 0 \]

Sezvo \(\cos(𝜃) = 0\), ipapo \(𝜃 = 90°\). Saka, kona iri pakati pemavector maviri i90 degrees.

Muenzaniso Mubvunzo 5: Kuverenga Chigadzirwa Chemuchinjikwa cheVectors

Dambudziko: Kana wapiwa mavector maviri muzvikamu zvitatu, 𝐀 = 𝐢 + 2𝐣 + 3𝐤 uye 𝐁 = 4𝐢 + 5𝐣 + 6𝐤, verenga vhector yechigadzirwa chakasiyana 𝐀 × 𝐁.

Kukurukurirana: Chibereko chekubatanidza mavector maviri muzvikamu zvitatu (𝐀 × 𝐁) ndeichi:

\[ 𝐀 × 𝐁 = \begin{vmatrix} 𝐢 & 𝐣 & 𝐤 \\ 1 & 2 & 3 \\ 4 & 5 & 6 \end{vmatrix} \]

\[ = 𝐢 (2 6 – 3 5) – 𝐣 (1 6 – 3 4) + 𝐤 (1 5 – 2 4) \]

\[ = 𝐢 (12 – 15) – 𝐣 (6 – 12) + 𝐤 (5 – 8) \]

\[ = 𝐢 (-3) – 𝐣 (-6) + 𝐤 (-3) \]

\[ = -3𝐢 + 6𝐣 – 3𝐤 \]

Saka, mhedzisiro yechigadzirwa chakasanganiswa 𝐀 × 𝐁 ndeye -3𝐢 + 6𝐣 – 3𝐤.

Mhedziso

Mufizikisi nemasvomhu, mavector inzira inobatsira zvikuru yekumiririra huwandu hune gwara uye hukuru. Nekunzwisisa maitiro ekuona zvikamu zvevector, kuverenga hukuru, kuwedzera mavector, uye kuverenga makona pakati pemavector uye zvigadzirwa zvakachinjika, tinogona kugadzirisa matambudziko akasiyana-siyana ane chekuita nemavector. Kukurukurirana kwemibvunzo yemuenzaniso iri pamusoro kunovavarira kubatsira kudzamisa kunzwisisa kwedu pfungwa iyi. Pakupedzisira, kugona kunzwisisa nekushanda nemavector hunyanzvi hunobatsira zvikuru muzvikamu zvakasiyana zvesainzi neinjiniya.