Tambayoyi da Tattaunawa game da Vectors Masu Girma Uku a Tsarin Daidaito na Cartesian
Vectors masu girma uku muhimmin ra'ayi ne a fannin lissafi da kimiyyar lissafi, waɗanda galibi ana amfani da su don wakiltar abubuwa ko abubuwan da suka faru a sararin samaniya mai girma uku. A cikin tsarin daidaitawar Cartesian, waɗannan vectors ana wakilta su da sassa uku, galibi ana nuna su azaman \( (x, y, z) \). Wannan labarin zai tattauna misalai da yawa na matsaloli da mafita da suka shafi vectors masu girma uku a cikin tsarin daidaitawar Cartesian.
Fahimtar Vectors Masu Girma Uku
Ana iya bayyana vector a cikin sarari mai girma uku kamar haka \(\mathbf{A} = (A_x, A_y, A_z)\), inda:
– \(A_x\) shine ɓangaren vector tare da axis ɗin x.
– \(A_y\) shine bangaren vector tare da axis ɗin y.
– \(A_z\) shine bangaren vector tare da axis ɗin z.
Tambayoyi da Tattaunawa Samfura
Tambaya ta 1: Aikin Ƙarin Vector
An ba da vector guda biyu, \(\mathbf{A} = (2, -3, 4)\) da \(\mathbf{B} = (-1, 5, 2)\). Lissafa jimlar waɗannan vector guda biyu.
Tattaunawa:
Ƙara vector guda biyu \(\mathbf{A}\) da \(\mathbf{B}\) ana yin su ne ta hanyar ƙara abubuwan da suka dace. Don haka, muna da:
\[
\mathbf{C} = \mathbf{A} + \mathbf{B} = (A_x + B_x, A_y + B_y, A_z + B_z)
\]
Sauya ƙimar vector da aka bayar:
\[
\mathbf{C} = (2 + (-1), -3 + 5, 4 + 2) = (1, 2, 6)
\]
Don haka, sakamakon ƙara vectors \(\mathbf{A}\) da \(\mathbf{B}\) shine \(\mathbf{C} = (1, 2, 6)\).
Tambaya ta 2: Aikin Rage Rage Vector
An ba da vector guda biyu, \(\mathbf{A} = (4, 1, -2)\) da \(\mathbf{B} = (5, -3, 6)\). Lissafa ragewar waɗannan vector guda biyu, wato \(\mathbf{A} – \mathbf{B}\).
Tattaunawa:
Rage vector guda biyu \(\mathbf{A}\) da \(\mathbf{B}\) ana yin su ne ta hanyar cire abubuwan da suka dace. Don haka, muna da:
\[
\mathbf{D} = \mathbf{A} - \mathbf{B} = (A_x – B_x, A_y – B_y, A_z – B_z)
\]
Sauya ƙimar vector da aka bayar:
\[
\mathbf{D} = (4 - 5, 1 - (-3), -2 - 6) = (-1, 4, -8)
\]
Don haka, sakamakon cire vectors \(\mathbf{A}\) da \(\mathbf{B}\) shine \(\mathbf{D} = (-1, 4, -8)\).
Tambaya ta 3: Aikin ninka sikelin
An ba da vector \(\mathbf{A} = (3, -2, 7)\) da scalar \(k = 4\). Lissafa samfurin scalar na waɗannan vectors.
Tattaunawa:
Ana yin ninka scalar \(k\) ta hanyar vector \(\mathbf{A}\) ta hanyar ninka kowane ɓangare na vector da wannan scalar. Don haka, muna da:
\[
\mathbf{E} = k \cdot \mathbf{A} = k \cdot (A_x, A_y, A_z) = (k \cdot A_x, k \cdot A_y, k \cdot A_z)
\]
Maye gurbin dabi'un da aka bayar:
\[
\mathbf{E} = 4 \cdot (3, -2, 7) = (4 \cdot 3, 4 \cdot -2, 4 \cdot 7) = (12, -8, 28)
\]
Don haka, sakamakon ninka scalar \(k\) ta hanyar vector \(\mathbf{A}\) shine \(\mathbf{E} = (12, -8, 28)\).
Tambaya ta 4: Tsawon Vector
Lissafa tsawon (girma) na vector \(\mathbf{A} = (1, 2, 2)\).
Tattaunawa:
Ana iya ƙididdige tsayi ko girman vector \(\mathbf{A} = (A_x, A_y, A_z)\) ta amfani da dabarar:
\[
|\mathbf{A}| = \sqrt{A_x^2 + A_y^2 + A_z^2}
\]
Maye gurbin dabi'un da aka bayar:
\[
|\mathbf{A}| = \sqrt{1^2 + 2^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3
\]
Don haka, tsawon vector \(\mathbf{A}\) shine 3.
Tambaya ta 5: Samfurin Dot
An ba da vector guda biyu, \(\mathbf{A} = (1, 0, -1)\) da \(\mathbf{B} = (2, 3, 4)\). Lissafa samfurin digo na waɗannan vector guda biyu.
Tattaunawa:
Ana yin samfurin digo na vector guda biyu \(\mathbf{A} = (A_x, A_y, A_z)\) da \(\mathbf{B} = (B_x, B_y, B_z)\) ta hanyar ninka abubuwan da suka dace sannan a ƙara su. Don haka, muna da:
\[
\mathbf{A} \cdot \mathbf{B} = A_x \cdot B_x + A_y \cdot B_y + A_z \cdot B_z
\]
Maye gurbin dabi'un da aka bayar:
\[
\mathbf{A} \cdot \mathbf{B} = (1 \cdot 2) + (0 \cdot 3) + (-1 \cdot 4) = 2 + 0 – 4 = -2
\]
Don haka, samfurin digo na vectors \(\mathbf{A}\) da \(\mathbf{B}\) shine -2.
Tambaya ta 6: Kayayyaki Masu Juyawa
An ba da vector guda biyu, \(\mathbf{A} = (1, 2, 3)\) da \(\mathbf{B} = (4, 5, 6)\). Lissafa samfurin giciye na waɗannan vector guda biyu.
Tattaunawa:
Ana yin samfurin giciye na vectors guda biyu \(\mathbf{A} = (A_x, A_y, A_z)\) da \(\mathbf{B} = (B_x, B_y, B_z)\) ta amfani da dabarar da ke ƙasa:
\[
\mathbf{A} \times \mathbf{B} = \left( (A_y \cdot B_z – A_z \cdot B_y), (A_z \cdot B_x – A_x \cdot B_z), (A_x \cdot B_y – A_y \cdot B_x) \right)
\]
Maye gurbin dabi'un da aka bayar:
\[
\mathbf{A} \times \mathbf{B} = \left( (2 \cdot 6 – 3 \cdot 5), (3 \cdot 4 – 1 \cdot 6), (1 \cdot 5 – 2 \cdot 4) \right) = (12 – 15, 12 – 6, 5 – 8) = (-3, 6, -3)
\]
Don haka, samfurin giciye na vectors \(\mathbf{A}\) da \(\mathbf{B}\) shine \(\mathbf{A} \times \mathbf{B} = (-3, 6, -3)\).
Kammalawa
Vectors masu girma uku a cikin tsarin daidaitawar Cartesian kayan aiki ne masu mahimmanci a fannoni daban-daban na kimiyya da injiniyanci. Ta hanyar misalai da tattaunawa da ke sama, mun ga yadda ake gudanar da ayyuka daban-daban na asali akan vectors, kamar ƙari, ragi, ninka sikelin, da samfuran digo da giciye. Fahimtar waɗannan ra'ayoyi mai ƙarfi ba wai kawai a cikin lissafi ba har ma a cikin aikace-aikacen aikace-aikace a cikin kimiyyar lissafi, injiniyanci, da kimiyyar kwamfuta.
