Question Number 135821 by mnjuly1970 last updated on 16/Mar/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:….{nice}\:\:\:…..\:\:\:{calculus}….\: \\ $$$$\:\:\:\:{prove}\:{that}\::: \\ $$$$\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\frac{{ln}\left(\mathrm{1}−{x}\right)}{\mathrm{1}−\sqrt{\mathrm{1}−{x}}}\right){dx}=\mathrm{4}\left(\mathrm{1}−\zeta\left(\mathrm{2}\right)\right) \\ $$$$ \\ $$ Answered by mathmax by abdo…
Question Number 135790 by benjo_mathlover last updated on 16/Mar/21 $${Find}\:{the}\:{component}\:{form}\:{of} \\ $$$${the}\:{vector}\:{that}\:{reprecents}\:{the} \\ $$$${velocity}\:{of}\:{an}\:{airplane}\:{descending} \\ $$$${at}\:{speed}\:{of}\:\mathrm{150}\:{miles}\:{per}\:{hour} \\ $$$${at}\:{angle}\:\mathrm{20}°\:{below}\:{the}\:{horizontal} \\ $$ Terms of Service Privacy Policy…
Question Number 4548 by Yozzii last updated on 07/Feb/16 Commented by Yozzii last updated on 07/Feb/16 $${In}\:{the}\:{diagram}\:{is}\:{a}\:{parallelogram}\:{ABCD} \\ $$$${with}\:{diagonal}\:{CB}. \\ $$$${E}\:{and}\:{F}\:{are}\:{the}\:{midpoints}\:{of}\:{CD}\:{and} \\ $$$${BD}\:{respectively}.\:{Using}\:{vectors},\:{prove} \\ $$$${that}\:{AE}\:{and}\:{AF}\:{trisect}\:{CB}.…
Question Number 135423 by benjo_mathlover last updated on 13/Mar/21 $${If}\:\overset{\rightarrow} {{a}}=\left(\mathrm{4},\mathrm{2},−\mathrm{1}\right),\:\overset{\rightarrow} {{b}}=\left({m},\mathrm{1},\mathrm{1}\right) \\ $$$$\overset{\rightarrow} {{c}}=\left(\bar {\mathrm{3}}−\mathrm{1},\mathrm{0}\right)\:{are}\:{three}\:{vectors} \\ $$$${then}\:{find}\:{the}\:{value}\:{of}\:{m}\:{such} \\ $$$${that}\:\overset{\rightarrow} {{a}},\overset{\rightarrow} {{b}}\:{and}\:\overset{\rightarrow} {{c}}\:{are}\:{coplanar}\:{and} \\ $$$${find}\:\overset{\rightarrow}…
Question Number 69620 by aseer imad last updated on 25/Sep/19 Commented by kaivan.ahmadi last updated on 26/Sep/19 $${b}\:{is}\:{answer}\:{since}\: \\ $$$$\frac{\mathrm{1}}{\mathrm{5}}\left(\mathrm{4},\mathrm{0},−\mathrm{3}\right).\left(\mathrm{3},−\mathrm{1},\mathrm{4}\right)=\frac{\mathrm{1}}{\mathrm{5}}\left(\mathrm{12}+\mathrm{0}−\mathrm{12}\right)=\mathrm{0}\Rightarrow\overset{\rightarrow} {{b}}\:{is} \\ $$$${perpendicular}\:{to}\:\mathrm{3}{i}−{j}+\mathrm{4}{k} \\ $$$${and}…
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Question Number 68591 by TawaTawa last updated on 14/Sep/19 Commented by kaivan.ahmadi last updated on 14/Sep/19 $$\mathrm{102}+\mathrm{2}{p}+\mathrm{3}{q}=\mathrm{0} \\ $$$$\mathrm{17}+\mathrm{3}{p}−\mathrm{4}{q}=\mathrm{0} \\ $$$$\Rightarrow \\ $$$$\begin{cases}{\mathrm{2}{p}+\mathrm{3}{q}=−\mathrm{102}}\\{\mathrm{3}{p}−\mathrm{4}{q}=−\mathrm{17}}\end{cases}\Rightarrow\begin{cases}{−\mathrm{6}{p}−\mathrm{9}{q}=\mathrm{306}}\\{\mathrm{6}{p}−\mathrm{8}{q}=−\mathrm{34}}\end{cases}\Rightarrow \\ $$$$−\mathrm{17}{q}=\mathrm{272}\Rightarrow{q}=−\mathrm{16}…
Question Number 133925 by bemath last updated on 25/Feb/21 $$\:\mathrm{Given}\:\mathrm{vector}\:\overset{\rightarrow} {{a}}\:=\:\hat {\mathrm{i}}−\mathrm{2}\hat {\mathrm{j}}+\hat {\mathrm{k}}\:,\: \\ $$$$\overset{\rightarrow} {{b}}=\:\mathrm{2}\hat {\mathrm{i}}+\hat {\mathrm{j}}−\mathrm{2}\hat {\mathrm{k}}\:,\:\overset{\rightarrow} {{c}}=−\hat {\mathrm{i}}+\mathrm{3}\hat {\mathrm{j}}−\hat {\mathrm{k}} \\…
Question Number 133857 by mnjuly1970 last updated on 24/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:…..#{advanced}\:\:\:\:……………\:\:\:{calculus}#….. \\ $$$$\:\:\:\:{prove}\:\:{that}\::::\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}^{\mathrm{2}} \left(\mathrm{1}−{x}\right)}{{x}}{dx}\overset{?} {=}\mathrm{2}\zeta\left(\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\:\:\overset{\mathrm{1}−{x}={t}} {=}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}^{\mathrm{2}} \left({t}\right)}{\mathrm{1}−{t}}{dt}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \underset{{n}=\mathrm{0}} {\overset{\infty}…