Vectors muhimmin ra'ayi ne a fannin kimiyyar lissafi, wanda ake amfani da shi wajen wakiltar adadi tare da girma da alkibla. A fannin kimiyyar lissafi, ana amfani da vectors sau da yawa don bayyana abubuwa daban-daban kamar ƙarfi, gudu, hanzari, da ƙari. Wannan labarin zai tattauna misalai da yawa na matsalolin vector na kimiyyar lissafi, tare da mafita da bayaninsu.
1. Ƙarin Vector da Ragewa
Misali Tambaya ta 1:
An bayar da vector guda biyu \(\mathbf{A}\) da \(\mathbf{B}\) kamar haka:
\[
\mathbf{A} = 3\mathbf{i} + 4\mathbf{j}
\]
\[
\mathbf{B} = -2\mathbf{i} + 5\mathbf{j}
\]
Lissafa:
1. \(\mathbf{A} + \mathbf{B}\)
2. \(\mathbf{A} – \mathbf{B}\)
Mafita:
Domin ƙara vector guda biyu, muna ƙara abubuwan da suka haɗa daban-daban.
1. \(\mathbf{A} + \mathbf{B}\):
\[
\mathbf{A} + \mathbf{B} = (3\mathbf{i} + 4\mathbf{j}) + (-2\mathbf{i} + 5\mathbf{j})
\]
\[
= (3 - 2) \mathbf{i} + (4 + 5) \mathbf{j}
\]
\[
= 1\mathbf{i} + 9\mathbf{j}
\]
\[
\mathbf{A} + \mathbf{B} = \mathbf{i} + 9\mathbf{j}
\]
2. \(\mathbf{A} – \mathbf{B}\):
\[
\mathbf{A} - \mathbf{B} = (3\mathbf{i} + 4\mathbf{j}) - (-2\mathbf{i} + 5\mathbf{j})
\]
\[
= (3 – (-2))\mathbf{i} + (4 – 5)\mathbf{j}
\]
\[
= (3 + 2) \mathbf{i} + (-1)\mathbf{j}
\]
\[
= 5\mathbf{i} – \mathbf{j}
\]
Don haka, sakamakon shine:
\[
\mathbf{A} - \mathbf{B} = 5\mathbf{i} - \mathbf{j}
\]
2. Yawan Girman Sikeli (Samfurin Dot)
Misali Tambaya ta 2:
An bayar da vector guda biyu \(\mathbf{C}\) da \(\mathbf{D}\) kamar haka:
\[
\mathbf{C} = 6\mathbf{i} + 2\mathbf{j}
\]
\[
\mathbf{D} = 3\mathbf{i} + 4\mathbf{j}
\]
Lissafin samfurin scalar (samfurin ɗigo) na \(\mathbf{C}\) da \(\mathbf{D}\).
Mafita:
Samfurin sikelin vector guda biyu \(\mathbf{C}\) da \(\mathbf{D}\) shine:
\[
\mathbf{C} \cdot \mathbf{D} = (6\mathbf{i} + 2\mathbf{j}) \cdot (3\mathbf{i} + 4\mathbf{j})
\]
\[
= 6 \cdot 3 + 2 \cdot 4
\]
\[
= 18 + 8
\]
\[
= 26
\]
Don haka, sakamakon samfurin scalar na \(\mathbf{C}\) da \(\mathbf{D}\) shine 26.
3. Kayayyaki Masu Juyawa
Misali Tambaya ta 3:
An bayar da vector guda biyu \(\mathbf{E}\) da \(\mathbf{F}\) kamar haka:
\[
\mathbf{E} = \mathbf{i} + 2\mathbf{j} + 3\mathbf{k}
\]
\[
\mathbf{F} = 4\mathbf{i} + 5\mathbf{j} + 6\mathbf{k}
\]
Lissafa samfurin giciye na \(\mathbf{E}\) da \(\mathbf{F}\).
Mafita:
Ana iya ƙididdige samfurin giciye na vectors guda biyu \(\mathbf{E}\) da \(\mathbf{F}\) ta amfani da matrix determinator:
\[
\mathbf{E} \times \mathbf{F} = \fara{vmatrix}
\mathbf{i} & \mathbf{j} & \mathbf{k} \\
1 da 2 da 3 \\
4&5&6
\end{vmatrix}
\]
Lissafa mai ƙayyade matrix:
\[
\mathbf{E} \times \mathbf{F} = \mathbf{i} (2 \cdot 6 – 3 \cdot 5) – \mathbf{j} (1 \cdot 6 – 3 \cdot 4) + \mathbf{k} (1 \cdot 5 – 4) \cdot 2.
\]
\[
= \mathbf{i} (12 – 15) – \mathbf{j} (6 – 12) + \mathbf{k} (5 – 8)
\]
\[
= \mathbf{i} (-3) - \mathbf{j} (-6) + \mathbf{k} (-3)
\]
\[
= -3\mathbf{i} + 6\mathbf{j} - 3\mathbf{k}
\]
Don haka, sakamakon haɗin gwiwar samfurin \(\mathbf{E}\) da \(\mathbf{F}\) shine:
\[
\mathbf{E} \times \mathbf{F} = -3\mathbf{i} + 6\mathbf{j} – 3\mathbf{k}
\]
4. Girman Vector
Misali Tambaya ta 4:
Idan aka ba da vector \(\mathbf{G} = 3\mathbf{i} – 4\mathbf{j}\). Lissafa girman (tsawon) vector \(\mathbf{G}\).
Mafita:
Ana iya ƙididdige girman vector \(\mathbf{G}\) ta amfani da dabarar:
\[
|\mathbf{G}| = \sqrt{(3)^2 + (-4)^2}
\]
\[
= \sqrt{9 + 16}
\]
\[
= \sqrt{25}
\]
\[
= 5
\]
Don haka, girman vector \(\mathbf{G}\) shine 5.
5. Ƙudurin Vector
Misali Tambaya ta 5:
Vektor ɗin \(\mathbf{H}\) yana da girman raka'a 10 kuma yana samar da kusurwar 30° tare da axis ɗin x. Kayyade abubuwan da ke cikin vektor ɗin \(\mathbf{H}\) akan axes ɗin x- da y.
Mafita:
Ana iya ƙididdige sassan vector \(\mathbf{H}\) akan x (\(\mathbf{H}_x\)) da y (\(\mathbf{H}_y\)) ta amfani da trigonometry:
\[
\mathbf{H}_x = |\mathbf{H}| \kos(\theta)
\]
\[
\mathbf{H}_y = |\mathbf{H}| \sin(\theta)
\]
Tare da \(|\mathbf{H}| = 10\) da \(\theta = 30°\):
\[
\mathbf{H}_x = 10 \cos(30°)
\]
\[
\mathbf{H}_y = 10 \sin(30°)
\]
Ƙimar \(\cos(30°) = \frac{\sqrt{3}}{2}\) da \(\sin(30°) = \frac{1}{2}\):
\[
\mathbf{H}_x = 10 \cdot \frac{\sqrt{3}}{2} = 5\sqrt{3}
\]
\[
\mathbf{H}_y = 10 \cdot \frac{1}{2} = 5
\]
Don haka, abubuwan da ke cikin vector \(\mathbf{H}\) sune:
\[
\mathbf{H}_x = 5\sqrt{3}
\]
\[
\mathbf{H}_y = 5
\]
Kammalawa
A cikin wannan labarin, mun tattauna misalai da dama da suka shafi vectors a fannin kimiyyar lissafi, tun daga ƙarawa da rage vector, ninkawa scalar da cross, zuwa girman vector da ƙuduri. Fahimtar ra'ayi da aikin vectors yana da mahimmanci a fannin kimiyyar lissafi domin ana iya bayyana abubuwa da yawa na halitta ta amfani da vectors. Da fatan waɗannan matsalolin misalai za su taimaka muku fahimtar ra'ayin vectors sosai.
