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Question Number 117543 by bemath last updated on 12/Oct/20

If vector a^→ +b^→ +c^→ =0 ∣a^→ ∣=7, ∣b^→ ∣=3 and ∣c^→ ∣=5 find the angle vector a^→ and c^→ ?

$$\mathrm{If}\:\mathrm{vector}\:\overset{\rightarrow} {\mathrm{a}}+\overset{\rightarrow} {\mathrm{b}}+\overset{\rightarrow} {\mathrm{c}}=\mathrm{0} \\ $$$$\mid\overset{\rightarrow} {\mathrm{a}}\mid=\mathrm{7},\:\mid\overset{\rightarrow} {\mathrm{b}}\mid=\mathrm{3}\:\mathrm{and}\:\mid\overset{\rightarrow} {\mathrm{c}}\mid=\mathrm{5} \\ $$$$\mathrm{find}\:\mathrm{the}\:\mathrm{angle}\:\mathrm{vector}\:\overset{\rightarrow} {\mathrm{a}}\:\mathrm{and}\:\overset{\rightarrow} {\mathrm{c}}\:? \\ $$

Answered by bobhans last updated on 12/Oct/20

let α=∡(a^→ ,c^→ ) by Cosine of law ⇒cos (π−α) = ((7^2 +5^2 −3^2 )/(2.7.5)) ⇒−cos α= ((65)/(70)) ; cos α = −((13)/(14)) ⇒α = cos^(−1) (−((13)/(14))) = 158.21°

$$\mathrm{let}\:\alpha=\measuredangle\left(\overset{\rightarrow} {\mathrm{a}},\overset{\rightarrow} {\mathrm{c}}\right) \\ $$$$\mathrm{by}\:\mathrm{Cosine}\:\mathrm{of}\:\mathrm{law} \\ $$$$\Rightarrow\mathrm{cos}\:\left(\pi−\alpha\right)\:=\:\frac{\mathrm{7}^{\mathrm{2}} +\mathrm{5}^{\mathrm{2}} −\mathrm{3}^{\mathrm{2}} }{\mathrm{2}.\mathrm{7}.\mathrm{5}} \\ $$$$\Rightarrow−\mathrm{cos}\:\alpha=\:\frac{\mathrm{65}}{\mathrm{70}}\:;\:\mathrm{cos}\:\alpha\:=\:−\frac{\mathrm{13}}{\mathrm{14}} \\ $$$$\Rightarrow\alpha\:=\:\mathrm{cos}^{−\mathrm{1}} \left(−\frac{\mathrm{13}}{\mathrm{14}}\right)\:=\:\mathrm{158}.\mathrm{21}° \\ $$

Answered by AbduraufKodiriy last updated on 12/Oct/20

a^→ +c^→ =−b^→ ⇒ ∣a^→ +c^→ ∣=∣b^→ ∣ ⇒ ⇒ (√(∣a^→ ∣^2 +2∙a^→ ∙c^→ +∣c^→ ∣^2 ))=3 (√(7^2 +2∙∣a^→ ∣∣b^→ ∣cos(𝛟)+5^2 ))=3 74+2∙7∙5cos(𝛟)=9 ⇒ 70cos(𝛟)=−65 ⇒ ⇒ cos(𝛟)=−((13)/(14)) ⇒ 𝛟=π−arccos(((13)/(14))) 𝛟 is angle vector a^→ and c^→

$$\overset{\rightarrow} {\boldsymbol{{a}}}+\overset{\rightarrow} {\boldsymbol{{c}}}=−\overset{\rightarrow} {\boldsymbol{{b}}}\:\Rightarrow\:\mid\overset{\rightarrow} {\boldsymbol{{a}}}+\overset{\rightarrow} {\boldsymbol{{c}}}\mid=\mid\overset{\rightarrow} {\boldsymbol{{b}}}\mid\:\Rightarrow\: \\ $$$$\Rightarrow\:\sqrt{\mid\overset{\rightarrow} {\boldsymbol{{a}}}\mid^{\mathrm{2}} +\mathrm{2}\centerdot\overset{\rightarrow} {\boldsymbol{{a}}}\centerdot\overset{\rightarrow} {\boldsymbol{{c}}}+\mid\overset{\rightarrow} {\boldsymbol{{c}}}\mid^{\mathrm{2}} }=\mathrm{3} \\ $$$$\sqrt{\mathrm{7}^{\mathrm{2}} +\mathrm{2}\centerdot\mid\overset{\rightarrow} {\boldsymbol{{a}}}\mid\mid\overset{\rightarrow} {\boldsymbol{{b}}}\mid\boldsymbol{{cos}}\left(\boldsymbol{\varphi}\right)+\mathrm{5}^{\mathrm{2}} }=\mathrm{3} \\ $$$$\mathrm{74}+\mathrm{2}\centerdot\mathrm{7}\centerdot\mathrm{5}\boldsymbol{{cos}}\left(\boldsymbol{\varphi}\right)=\mathrm{9}\:\Rightarrow\:\mathrm{70}\boldsymbol{{cos}}\left(\boldsymbol{\varphi}\right)=−\mathrm{65}\:\Rightarrow \\ $$$$\Rightarrow\:\boldsymbol{{cos}}\left(\boldsymbol{\varphi}\right)=−\frac{\mathrm{13}}{\mathrm{14}}\:\Rightarrow\:\boldsymbol{\varphi}=\pi−\boldsymbol{{arccos}}\left(\frac{\mathrm{13}}{\mathrm{14}}\right) \\ $$$$\boldsymbol{\varphi}\:\boldsymbol{{is}}\:\boldsymbol{{angle}}\:\boldsymbol{{vector}}\:\overset{\rightarrow} {\boldsymbol{{a}}}\:\boldsymbol{{and}}\:\overset{\rightarrow} {\boldsymbol{{c}}} \\ $$

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