Misalan Tambayoyi da Tattaunawa kan Fassarar Lissafi
Fassara wani canji ne na geometric wanda ke motsa kowane wuri a kan jirgin sama wani takamaiman nisa a cikin wani takamaiman alkibla. A cikin lissafi, ana amfani da fassara sau da yawa don canza abu zuwa alkibla ɗaya ba tare da canza siffarsa ko yanayinsa ba. A cikin wannan labarin, za mu tattauna misalai da dama na matsaloli da tattaunawa da suka shafi fassarar lissafi don taimaka wa masu karatu su fahimci wannan ra'ayi sosai.
Ka'idojin Asali na Fassara
Ana iya bayyana fassarar a cikin daidaitawa masu girma biyu ta amfani da bayanin vector. Idan an fassara maki A(x, y) ta hanyar vector \((a, b)\), to za a iya ƙididdige maki da ya fito daga A' (\(x'\), \(y'\)) ta amfani da dabarar:
\[ x' = x + a \]
\[y' = y + b \]
Ina:
– \( (x, y) \) shine farkon daidaitawa,
– \((a, b) \) shine vector na fassara,
– \( (x', y') \) sune daidaitattun sakamakon fassarar.
Tambayoyi da Tattaunawa Samfura
Ga wasu misalan tambayoyi game da fassarar lissafi da tattaunawarsu:
Misali Tambaya ta 1:
Tambaya:
Maki A yana kan daidaitawa (3, 4). Fassara ma'aunin A da vector \( (5, -2) \). Ƙayyade sabbin daidaitawa na ma'aunin A.
Tattaunawa:
An sani:
\[ \text{Asalin daidaitawar maki A} = (3, 4) \]
\[ \text{Fassara vector} = (5, -2) \]
Yi amfani da dabarar fassara:
\[ x' = x + a \]
\[y' = y + b \]
Maye gurbin dabi'un:
\[ x' = 3 + 5 = 8 \]
\[y' = 4 + (-2) = 2 \]
Don haka, sabbin daidaitawar maki A bayan fassara sune \( (8, 2) \).
Misali Tambaya ta 2:
Tambaya:
Alwatika tana da daidaitattun daidaiton ...
Tattaunawa:
An san daidaitattun kusurwar alwatika da kuma vector na fassara.
Daidaito na maki A':
\[ x' = 1 + (-2) = -1 \]
\[y' = 2 + 4 = 6 \]
Sai, \( A' = (-1, 6) \).
Daidaito na maki B':
\[ x' = 3 + (-2) = 1 \]
\[y' = 5 + 4 = 9 \]
Sai, \( B' = (1, 9) \).
Daidaito na ma'auni na C':
\[ x' = 6 + (-2) = 4 \]
\[y' = 1 + 4 = 5 \]
Sannan, \( C' = (4, 5) \).
Don haka, bayan fassarar, daidaitattun ma'aunin kusurwar sabon alwatika sune \( A'(-1, 6) \), \( B'(1, 9) \), da \( C'(4, 5) \).
Misali Tambaya ta 3:
Tambaya:
An bayar da maki P a coordinates \( (-3, 0) \). Kayyade sakamakon fassarar maki P ta hanyar vector \( (7, -5) \).
Tattaunawa:
Ganin yadda aka yi amfani da daidaitattun P da kuma vector na fassara.
Yi amfani da dabarar fassara:
\[ x' = x + a \]
\[y' = y + b \]
Maye gurbin dabi'un:
\[ x' = -3 + 7 = 4 \]
\[y' = 0 + (-5) = -5 \]
Don haka, daidaitattun sakamakon fassarar maki P sune \( (4, -5) \).
Misali Tambaya ta 4:
Tambaya:
Ma'anar Q tana nan a ma'aunin \( (4, -3) \). Idan an fassara ma'aunin Q ta yadda sabbin ma'aunin za su zama \( (9, 1) \), a tantance ma'aunin fassarar da aka yi amfani da shi.
Tattaunawa:
An sani:
\[ \text{Hanyoyin farko} = (4, -3) \]
\[ \text{Sakamakon daidaitawa} = (9, 1) \]
Yi amfani da dabarar fassara don nemo vector \( (a, b) \):
\[ x' = x + a \]
\[y' = y + b \]
An san sakamakon fassarar:
\[ 9 = 4 + a \]
\[ 1 = -3 + b \]
Saboda haka, ana iya ƙididdige vector na fassara kamar haka:
\[ a = 9 – 4 = 5 \]
\[ b = 1 + 3 = 4 \]
Don haka, hanyar fassara da aka yi amfani da ita ita ce \( (5, 4) \).
Misali Tambaya ta 5:
Tambaya:
ABCD mai kusurwa huɗu yana da maki kusurwa \( A(1, 2) \), \( B(1, 5) \), \( C(4, 5) \), da \( D(4, 2) \). Fassara quadrilateral tare da vector \( (3, -1) \). Ƙayyade sabbin daidaitawa na quadrilateral ABCD.
Tattaunawa:
An sani:
\[ \text{Fassara vector} = (3, -1) \]
Daidaito na maki A':
\[ x' = 1 + 3 = 4 \]
\[y' = 2 + (-1) = 1 \]
Sai, \( A' = (4, 1) \).
Daidaito na maki B':
\[ x' = 1 + 3 = 4 \]
\[y' = 5 + (-1) = 4 \]
Sai, \( B' = (4, 4) \).
Daidaito na ma'auni na C':
\[ x' = 4 + 3 = 7 \]
\[y' = 5 + (-1) = 4 \]
Sannan, \( C' = (7, 4) \).
Daidaito na maki D':
\[ x' = 4 + 3 = 7 \]
\[y' = 2 + (-1) = 1 \]
Sai, \( D' = (7, 1) \).
Don haka, sabbin daidaitattun ABCD mai kusurwa huɗu bayan fassara sune \( A'(4, 1) \), \( B'(4, 4) \), \( C'(7, 4) \), da \( D'(7, 1) \).
Kammalawa
Fassara wani sauyi ne mai sauƙi amma mai mahimmanci a fannin lissafi. Kwarewar wannan dabarar tana ba mu damar yin ayyuka daban-daban na lissafi, kamar canza abubuwa ba tare da canza siffarsu ko girmansu ba.
Ta hanyar fahimtar manufar fassara ta hanyar misalan matsaloli daban-daban da aka tattauna a sama, ana fatan masu karatu za su iya fahimtar da kuma amfani da wannan ra'ayi a cikin matsaloli daban-daban da kuma a rayuwa ta ainihi. Fassara tana da amfani ba kawai a lissafi ba har ma a wasu fannoni daban-daban, gami da kimiyyar lissafi, zane-zanen kwamfuta, da ƙira.
