Tusaale Su'aal Dood ah oo ku saabsan Vektor-ka Cutubka ee Vektor-ka
Pendahuluan
Xisaabta iyo fiisigiska, vektarrada waa walxo aasaasi ah oo matalaya baaxadda iyo jihada. Vektarrada waxaa badanaa loo isticmaalaa in lagu qeexo ifafaale kala duwan sida xawaaraha, xoogga, iyo barokaca booska laba ama saddex-cabbir ah. Hal fikrad oo muhiim ah oo la xiriirta vektarrada waa vektarrada cutubka. Maqaalkani wuxuu ka hadli doonaa qeexidda vektarrada cutubka, sida loo xisaabiyo, wuxuuna bixin doonaa dhowr tusaale oo dhibaatooyin iyo xalal ah.
Fahmidda Vektorrada Cutubka
Vektor cutub waa vektor leh cabbir hal cutub ah iyo jiho la mid ah vektor-kii asalka ahaa. Vektor-yada cutubyada waxaa badanaa loo isticmaalaa in lagu fududeeyo falanqaynta sababtoo ah baaxaddoodu had iyo jeer waa hal, taasoo u oggolaanaysa diiradda koowaad inay ku socoto jihadooda. Si aan vektor ugu beddelno vektor cutub, waa inaan qayb kasta oo ka mid ah qaybaheeda u qaybinnaa cabbirka vektor-ka.
Xisaab ahaan, haddii \( \mathbf{v} \) uu yahay vektor, markaa vektor-kiisa cutubka \( \mathbf{\hat{v}} \) waxaa loo sheegi karaa sidan:
\[
\mathbf{\hat{v}} = \frac{\mathbf{v}}{\|\mathbf{v}\|}
\]
halkaas oo \( \|\mathbf{v}\| \) uu yahay baaxadda ama dhererka vector-ka \( \mathbf{v} \).
Xisaabinta Cabbirka Vektor-ka
Baaxadda vektor \( \mathbf{v} \) ee booska laba-geesoodka ah oo leh qaybaha \( (v_x, v_y) \) waxaa lagu xisaabin karaa iyadoo la adeegsanayo qaacidada:
\[
\|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2}
\]
Dhanka kale, vektorrada ku jira meel saddex-cabbir ah oo leh qaybaha \( (v_x, v_y, v_z) \), cabbirka waxaa lagu xisaabiyaa iyadoo la adeegsanayo qaacidada:
\[
\|\mathbf{v}\| = \sqrt{v_x^2 + v_y^2 + v_z^2}
\]
Su'aalo iyo Doodo Tusaale ah
Si aan u caddayno fikradda vectors-ka cutubyada, aan eegno tusaalooyin su'aalo ah iyo doodahooda.
Su'aal Tusaale 1aad
Su'aal: La siiyay vektor \( \mathbf{a} = (3, 4) \). Go'aami vektor-ka cutubka ee vektor-ka \( \mathbf{a} \).
Dood:
1. Go'aami qaybaha vektorka \( \mathbf{a} \):
\( a_x = 3 \), \( a_y = 4 \)
2. Xisaabi baaxadda vektorka \( \mathbf{a} \):
\[
\|\mathbf{a}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
3. Xisaabi vektor-ka cutubka adigoo qayb kasta oo ka mid ah vektor-ka u qaybinaya baaxaddiisa:
\[
\mathbf{\hat{a}} = \left( \frac{3}{5}, \frac{4}{5} \right) = \left( 0.6, 0.8 \right)
\]
Markaa, halbeegga halbeegga ee \( \mathbf{a} \) waa \( (0.6, 0.8) \).
Su'aal Tusaale 2aad
Su'aal: La siiyay vektor \( \mathbf{b} = (1, -2, 2) \). Go'aami vektor-ka cutubka ee vektor-ka \( \mathbf{b} \).
Dood:
1. Go'aami qaybaha vektorka \( \mathbf{b} \):
\( b_x = 1 \), \( b_y = -2 \), \( b_z = 2 \)
2. Xisaabi baaxadda vektorka \( \mathbf{b} \):
\[
\|\mathbf{b}\| = \sqrt{1^2 + (-2)^2 + 2^2} = \sqrt{1 + 4 + 4} = \sqrt{9} = 3
\]
3. Xisaabi vektor-ka cutubka adigoo qayb kasta oo ka mid ah vektor-ka u qaybinaya baaxaddiisa:
\[
\mathbf{\hat{b}} = \left( \frac{1}{3}, \frac{-2}{3}, \frac{2}{3} \right) \approx \left( 0.333, -0.667, 0.667 \right)
\]
Markaa, vektor-ka cutubka ee \( \mathbf{b} \) waa \( \left( 0.333, -0.667, 0.667 \right) \).
Su'aal Tusaale 3aad
Su'aal: Marka la eego vektorka \( \mathbf{c} = (-7, 24) \). Go'aami vektorka cutubka ee vektorka \( \mathbf{c} \).
Dood:
1. Go'aami qaybaha vektorka \( \mathbf{c} \):
\( c_x = -7 \), \( c_y = 24 \)
2. Xisaabi baaxadda vektorka \( \mathbf{c} \):
\[
\|\mathbf{c}\| = \sqrt{(-7)^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25
\]
3. Xisaabi vektor-ka cutubka adigoo qayb kasta oo ka mid ah vektor-ka u qaybinaya baaxaddiisa:
\[
\mathbf{\hat{c}} = \left( \frac{-7}{25}, \frac{24}{25} \right) = \left( -0.28, 0.96 \right)
\]
Markaa, vektor-ka cutubka ee \( \mathbf{c} \) waa \( (-0.28, 0.96) \).
Su'aal Tusaale 4aad
Su'aal: Haddii vektor \( \mathbf{d} = (6, 8, 0) \), go'aami vektor-ka cutubka vektor \( \mathbf{d} \).
Dood:
1. Go'aami qaybaha vektorka \( \mathbf{d} \):
\( d_x = 6 \), \( d_y = 8 \), \( d_z = 0 \)
2. Xisaabi baaxadda vektorka \( \mathbf{d} \):
\[
\|\mathbf{d}\| = \sqrt{6^2 + 8^2 + 0^2} = \sqrt{36 + 64 + 0} = \sqrt{100} = 10
\]
3. Xisaabi vektor-ka cutubka adigoo qayb kasta oo ka mid ah vektor-ka u qaybinaya baaxaddiisa:
\[
\mathbf{\hat{d}} = \left( \frac{6}{10}, \frac{8}{10}, \frac{0}{10} \right) = \left( 0.6, 0.8, 0 \right)
\]
Markaa, vektor-ka cutubka ee \( \mathbf{d} \) waa \( (0.6, 0.8, 0) \).
Xiritaanka
Doodda iyo tusaalooyinka kor ku xusan awgeed, waxaan fahmi karnaa in xisaabinta vektor-ka cutubku ay u baahan tahay xisaabinta baaxadda vektor-ka ka dibna loo qaybiyo qaybaha vektor-ka cabbirkaas. Vektor-yada cutubyadu aad bay waxtar ugu leeyihiin codsiyada kala duwan sida caadiyeynta vektor-ka ee sawirada kombiyuutarka, falanqaynta xoogga ee fiisigiska, iyo meelo kale oo badan. Marka aan fahanno fikraddan, waa inaan si fudud u maareynaa dhibaatooyinka ku lug leh vektor-yada.
