Tambayoyi da Tattaunawa game da Saurin Girma ta hanyar Vectors
Pendahuluan
A fannin lissafi da kimiyyar lissafi, ninka sikelin ta hanyar vector aiki ne mai mahimmanci kuma ana yawan amfani da shi. Wannan ninka yana da mahimmanci don haɓaka ƙarin ra'ayoyi masu rikitarwa a cikin nazarin lissafi, makanikai, da vector. Wannan labarin yana da nufin bayyana manufar ninka sikelin ta hanyar vector kuma yana ba da misalai da tattaunawa don fayyace fahimta.
Fahimtar Saurin Girman Scalar tare da Vectors
Sauyawa scalar ta hanyar vector shine aikin da ake ninka scalar (lamba ɗaya) da kowace ɓangaren vector. Sakamakon wannan aikin shine sabon vector mai alkibla iri ɗaya da vector na asali amma tare da girma wanda scalar ya canza. Gabaɗaya, idan muna da vector \(\mathbf{v} = (v_1, v_2, v_3)\) da scalar \(k\), to samfurin su \(k \mathbf{v}\) shine:
\[
k \mathbf{v} = (k v_1, k v_2, k v_3)
\]
Tambayoyi da Tattaunawa Samfura
Tambaya ta 1
A ce akwai vector \(\mathbf{v} = (3, -4, 5)\) da scalar \(k = 2\). A lissafta samfurin scalar da vector.
Tattaunawa ta 1
Amfani da ma'anar ninka scalar ta hanyar vector:
\[
k \mathbf{v} = 2 \cdot (3, -4, 5)
\]
Matakan lissafi sune kamar haka:
\[
k \mathbf{v} = (2 \cdot 3, 2 \cdot -4, 2 \cdot 5)
\]
\[
k \mathbf{v} = (6, -8, 10)
\]
Don haka, samfurin scalar \(2\) ta hanyar vector \((3, -4, 5)\) shine \((6, -8, 10)\).
Tambaya ta 2
Idan akwai vector \(\mathbf{w} = (-1, 0, 7)\) da scalar \(k = -3\), ƙayyade samfurin scalar.
Tattaunawa ta 2
Amfani da dabarar iri ɗaya kamar yadda aka yi a baya:
\[
k \mathbf{w} = -3 \cdot (-1, 0, 7)
\]
Matakan lissafi sune kamar haka:
\[
k \mathbf{w} = (-3 \cdot -1, -3 \cdot 0, -3 \cdot 7)
\]
\[
k \mathbf{w} = (3, 0, -21)
\]
Samfurin scalar \(-3\) ta hanyar vector \(-1, 0, 7)\) shine \((3, 0, -21)\).
Tambaya ta 3
Akwai vector \(\mathbf{u} = (2, -1, 4)\). Idan vector ya ninka ta hanyar scalar \(\frac{1}{2}\), tantance sakamakon ninkawa.
Tattaunawa ta 3
Amfani da wannan dabarar:
\[
k \mathbf{u} = \frac{1}{2} \cdot (2, -1, 4)
\]
Matakan lissafi sune kamar haka:
\[
k \mathbf{u} = \left(\frac{1}{2} \cdot 2, \frac{1}{2} \cdot -1, \frac{1}{2} \cdot 4\dama)
\]
\[
k \mathbf{u} = (1, -0.5, 2)
\]
Don haka, samfurin scalar \(\frac{1}{2}\) ta hanyar vector \((2, -1, 4)\) shine \((1, -0.5, 2)\).
Tambaya ta 4
An ba da vector \(\mathbf{a} = (6, 8, -3)\) da scalar \(k = 0\). Nemo samfurin su.
Tattaunawa ta 4
Ta hanyar amfani da dabarar ninka scalar tare da vector:
\[
k \mathbf{a} = 0 \cdot (6, 8, -3)
\]
Matakan lissafi sune kamar haka:
\[
k \mathbf{a} = (0 \cdot 6, 0 \cdot 8, 0 \cdot -3)
\]
\[
k \mathbf{a} = (0, 0, 0)
\]
Samfurin scalar \(0\) ta hanyar vector \((6, 8, -3)\) shine \((0, 0, 0)\). Wannan yana nuna cewa ninka vector ta hanyar scalar \(0\) zai samar da sifili vector.
Tambaya ta 5
A ce akwai vector guda biyu \(\mathbf{b} = (7, -2, 3)\) da \(\mathbf{c} = (-5, 4, 6)\). Ka tantance samfurin scalar na \(4\) tare da jimlar vector guda biyu.
Tattaunawa ta 5
Mataki na farko shine a ƙara vectors guda biyu:
\[
\mathbf{b} + \mathbf{c} = (7, -2, 3) + (-5, 4, 6)
\]
Ana yin ƙarin vector ta hanyar ƙara abubuwan da suka dace:
\[
\mathbf{b} + \mathbf{c} = (7 + (-5), -2 + 4, 3 + 6)
\]
\[
\mathbf{b} + \mathbf{c} = (2, 2, 9)
\]
Mataki na gaba, ninka sakamakon da scalar \(4\):
\[
4 (\mathbf{b} + \mathbf{c}) = 4 \cdot (2, 2, 9)
\]
Matakan lissafi sune:
\[
4 (\mathbf{b} + \mathbf{c}) = (4 \cdot 2, 4 \cdot 2, 4 \cdot 9)
\]
\[
4 (\mathbf{b} + \mathbf{c}) = (8, 8, 36)
\]
Don haka, samfurin scalar na \(4\) da jimlar vectors guda biyu shine \((8, 8, 36)\).
Kammalawa
Ninninka sikelin da vector aiki ne mai sauƙi amma mai mahimmanci a fannoni da yawa na kimiyya. Ta hanyar ninka sikelin da kowane ɓangare na vector, za mu iya canza girman vector cikin sauƙi ba tare da canza alkiblarsa ba. Wannan labarin ya bayyana manufar kuma ya ba da misalai da mafita don fayyace yadda wannan aikin ke aiki. Fahimtar wannan aikin na asali zai iya sauƙaƙa wa mutum ya ƙware a fannoni masu ci gaba a fannin lissafi da kimiyyar lissafi.
Ana fatan cewa ta hanyar wannan labarin da tambayoyin misalai, masu karatu za su iya samun fahimtar yawan scalar ta hanyar vectors, kuma za su iya amfani da shi a cikin yanayi da matsaloli na gaske.
