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Question Number 141129 Answers: 1 Comments: 0

For what values of λ are the vectors λi^ + 2j^ + k^ , 3i^ +4j^ +λk^ , j^ + k^ coplanar

$${For}\:{what}\:{values}\:{of}\:\lambda\:{are}\:{the} \\ $$$${vectors}\:\lambda\hat {{i}}\:+\:\mathrm{2}\hat {{j}}\:+\:\hat {{k}}\:,\:\mathrm{3}\hat {{i}}\:+\mathrm{4}\hat {{j}}\:+\lambda\hat {{k}}\: \\ $$$$,\:\hat {{j}}\:+\:\hat {{k}}\:\:{coplanar}\: \\ $$

Question Number 140690 Answers: 1 Comments: 0

Vector let L_1 = AC where A=(2,−1,3) and C=(1,0,−5)and let L_2 = BD where B=(1,3,0) and D=(3,−4,1). Determine the distance between L_1 and L_2 .

$$\:\mathrm{Vector}\: \\ $$$$\mathrm{let}\:\mathcal{L}_{\mathrm{1}} =\:\mathrm{AC}\:\mathrm{where}\:\mathrm{A}=\left(\mathrm{2},−\mathrm{1},\mathrm{3}\right)\:\mathrm{and} \\ $$$$\mathrm{C}=\left(\mathrm{1},\mathrm{0},−\mathrm{5}\right)\mathrm{and}\:\mathrm{let}\:\mathcal{L}_{\mathrm{2}} =\:\mathrm{BD}\:\mathrm{where} \\ $$$$\mathrm{B}=\left(\mathrm{1},\mathrm{3},\mathrm{0}\right)\:\mathrm{and}\:\mathrm{D}=\left(\mathrm{3},−\mathrm{4},\mathrm{1}\right).\:\mathrm{Determine} \\ $$$$\mathrm{the}\:\mathrm{distance}\:\mathrm{between}\:\mathcal{L}_{\mathrm{1}} \:\mathrm{and}\:\mathcal{L}_{\mathrm{2}} . \\ $$

Question Number 140666 Answers: 1 Comments: 0

Find the coordinates of (−2,0) when the axes are rotated counterclockwise through the angle arcsin (4/5).

$${Find}\:{the}\:{coordinates}\:{of}\:\left(−\mathrm{2},\mathrm{0}\right)\: \\ $$$${when}\:{the}\:{axes}\:{are}\:{rotated}\:{counterclockwise} \\ $$$${through}\:{the}\:{angle}\:\mathrm{arcsin}\:\frac{\mathrm{4}}{\mathrm{5}}. \\ $$

Question Number 140502 Answers: 0 Comments: 0

If three vector a^→ , b^→ and c^→ are such that a^→ ≠ 0 and a^→ ×b^→ = 2(a^→ ×c^→ ) ,∣a^→ ∣ = ∣c^→ ∣ = 1 , ∣b^→ ∣ = 4 and the angle between b^→ and c^→ is cos^(−1) ((1/4)), then b^→ −2c^→ = λ a^→ , where λ =?

Question Number 140176 Answers: 0 Comments: 0

Let V be a vector space of polynomials p(x)= a+bx+cx^2 with real coefficients a,b and c. Define an inner product on V by (p,q)=(1/2)∫_(−1) ^1 p(x)q(x) dx . (a) Find a orthonormal basis for V consisting of polynomials φ_o (x) , φ_1 (x) and φ_2 (x) having degree 0,1 and 2 respectively.

$$\mathrm{Let}\:\mathrm{V}\:\mathrm{be}\:\mathrm{a}\:\mathrm{vector}\:\mathrm{space}\:\mathrm{of}\:\mathrm{polynomials} \\ $$$$\mathrm{p}\left(\mathrm{x}\right)=\:\mathrm{a}+\mathrm{bx}+\mathrm{cx}^{\mathrm{2}} \:\mathrm{with}\:\mathrm{real}\:\mathrm{coefficients} \\ $$$$\mathrm{a},\mathrm{b}\:\mathrm{and}\:\mathrm{c}.\:\mathrm{Define}\:\mathrm{an}\:\mathrm{inner}\:\mathrm{product}\:\mathrm{on}\:\mathrm{V} \\ $$$$\mathrm{by}\:\left(\mathrm{p},\mathrm{q}\right)=\frac{\mathrm{1}}{\mathrm{2}}\underset{−\mathrm{1}} {\overset{\mathrm{1}} {\int}}\mathrm{p}\left(\mathrm{x}\right)\mathrm{q}\left(\mathrm{x}\right)\:\mathrm{dx}\:. \\ $$$$\left(\mathrm{a}\right)\:\mathrm{Find}\:\mathrm{a}\:\mathrm{orthonormal}\:\mathrm{basis}\:\mathrm{for}\:\mathrm{V}\:\mathrm{consisting} \\ $$$$\mathrm{of}\:\mathrm{polynomials}\:\phi_{\mathrm{o}} \left(\mathrm{x}\right)\:,\:\phi_{\mathrm{1}} \left(\mathrm{x}\right)\:\mathrm{and}\:\phi_{\mathrm{2}} \left(\mathrm{x}\right) \\ $$$$\mathrm{having}\:\mathrm{degree}\:\mathrm{0},\mathrm{1}\:\mathrm{and}\:\mathrm{2}\:\mathrm{respectively}. \\ $$$$ \\ $$

Question Number 140053 Answers: 0 Comments: 3

Question Number 139992 Answers: 0 Comments: 1

Question Number 139455 Answers: 1 Comments: 0

# calculus# evaluate: 𝛗:=Σ_(k=1) ^∞ (((−1)^(k−1) Γ ((k/2)))/(k Γ(((k+1)/2)))) =?

$$\:\:\:\:\:\:\:#\:{calculus}# \\ $$$$\:\:{evaluate}: \\ $$$$\:\:\boldsymbol{\phi}:=\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{k}−\mathrm{1}} \:\Gamma\:\left(\frac{{k}}{\mathrm{2}}\right)}{{k}\:\Gamma\left(\frac{{k}+\mathrm{1}}{\mathrm{2}}\right)}\:=? \\ $$

Question Number 138976 Answers: 1 Comments: 0

Question Number 138366 Answers: 2 Comments: 0

If x^2 +x^(−2) =(√(2+(√(2+(√2))))) x^(16) +x^(−16) =? Any help

$$\boldsymbol{\mathrm{If}}\:\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{x}}^{−\mathrm{2}} =\sqrt{\mathrm{2}+\sqrt{\mathrm{2}+\sqrt{\mathrm{2}}}} \\ $$$$\boldsymbol{\mathrm{x}}^{\mathrm{16}} +\boldsymbol{\mathrm{x}}^{−\mathrm{16}} =? \\ $$$$\boldsymbol{\mathrm{Any}}\:\boldsymbol{\mathrm{help}} \\ $$

Question Number 137970 Answers: 1 Comments: 0

Prove that ▽^2 𝛗=−4𝛑Gρ φ=Potential of Gravitational field ρ=Density G=Universal Gravitational Constant

$${Prove}\:{that}\: \\ $$$$\bigtriangledown^{\mathrm{2}} \boldsymbol{\phi}=−\mathrm{4}\boldsymbol{\pi{G}}\rho\:\: \\ $$$$\phi={Potential}\:{of}\:{Gravitational}\:{field} \\ $$$$\rho={Density}\:\:\:\boldsymbol{{G}}={Universal}\:{Gravitational}\:{Constant} \\ $$

Question Number 136300 Answers: 3 Comments: 0

Let vector a^→ , b^→ and c^→ such that ∣a^→ ∣=∣b^→ ∣=((∣c^→ ∣)/2) and a^→ ×(a^→ ×c^→ )+b^→ =0 find the acute angle between a^→ and c^→ .

Question Number 135965 Answers: 0 Comments: 2

Vector Three vectors satisfy a.b = b.c = c.a = -1 and a + b + c = 0. What is the magnitude of vector a, b , and c?

$${Vector} \\ $$Three vectors satisfy a.b = b.c = c.a = -1 and a + b + c = 0. What is the magnitude of vector a, b , and c?

Question Number 135821 Answers: 4 Comments: 0

....nice ..... calculus.... prove that :: 𝛗=∫_0 ^( 1) (((ln(1−x))/(1−(√(1−x)))))dx=4(1−ζ(2))

$$\:\:\:\:\:\:\:\:\:\:\:\:\:....{nice}\:\:\:.....\:\:\:{calculus}....\: \\ $$$$\:\:\:\:{prove}\:{that}\::: \\ $$$$\:\:\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\frac{{ln}\left(\mathrm{1}−{x}\right)}{\mathrm{1}−\sqrt{\mathrm{1}−{x}}}\right){dx}=\mathrm{4}\left(\mathrm{1}−\zeta\left(\mathrm{2}\right)\right) \\ $$$$ \\ $$

Question Number 135790 Answers: 0 Comments: 0

Find the component form of the vector that reprecents the velocity of an airplane descending at speed of 150 miles per hour at angle 20° below the horizontal

$${Find}\:{the}\:{component}\:{form}\:{of} \\ $$$${the}\:{vector}\:{that}\:{reprecents}\:{the} \\ $$$${velocity}\:{of}\:{an}\:{airplane}\:{descending} \\ $$$${at}\:{speed}\:{of}\:\mathrm{150}\:{miles}\:{per}\:{hour} \\ $$$${at}\:{angle}\:\mathrm{20}°\:{below}\:{the}\:{horizontal} \\ $$

Question Number 135423 Answers: 1 Comments: 0

If a^→ =(4,2,−1), b^→ =(m,1,1) c^→ =(3^ −1,0) are three vectors then find the value of m such that a^→ ,b^→ and c^→ are coplanar and find a^→ ×(b^→ ×c^→ ).

Question Number 133925 Answers: 2 Comments: 0

Given vector a^→ = i^ −2j^ +k^ , b^→ = 2i^ +j^ −2k^ , c^→ =−i^ +3j^ −k^ and d^→ = 2j^ −2k^ . Find the value of (a^→ ×b^→ )×(c^→ ×d^→ ).

Question Number 133857 Answers: 2 Comments: 0

.....#advanced ............... calculus#..... prove that ::: 𝛗=∫_0 ^( 1) ((ln^2 (1−x))/x)dx=^? 2ζ(3) =^(1−x=t) ∫_0 ^( 1) ((ln^2 (t))/(1−t))dt=∫_0 ^( 1) Σ_(n=0) ^∞ ln^2 (t).t^n dt =Σ_(n=0) ^∞ {[(t^(n+1) /(n+1))ln^2 (t)]_0 ^1 −(2/(n+1))∫_0 ^( 1) t^n ln(t) =−2Σ_(n=0) ^∞ (1/(n+1)){[(t^(n+1) /(n+1))ln(t)]_0 ^1 −(1/(n+1))∫_0 ^( 1) t^n dt} =2Σ_(n=0) ^∞ (1/((n+1)^3 ))=2Σ_(n=1) ^∞ (1/n^3 )=2ζ(3) .................... 𝛗=2ζ(3) .................... ..........m.n.july.1970.........

$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:.....#{advanced}\:\:\:\:...............\:\:\:{calculus}#..... \\ $$$$\:\:\:\:{prove}\:\:{that}\::::\:\:\boldsymbol{\phi}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}^{\mathrm{2}} \left(\mathrm{1}−{x}\right)}{{x}}{dx}\overset{?} {=}\mathrm{2}\zeta\left(\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\:\:\overset{\mathrm{1}−{x}={t}} {=}\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}^{\mathrm{2}} \left({t}\right)}{\mathrm{1}−{t}}{dt}=\int_{\mathrm{0}} ^{\:\mathrm{1}} \underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}{ln}^{\mathrm{2}} \left({t}\right).{t}^{{n}} {dt} \\ $$$$\:\:\:\:\:=\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\left\{\left[\frac{{t}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}{ln}^{\mathrm{2}} \left({t}\right)\right]_{\mathrm{0}} ^{\mathrm{1}} −\frac{\mathrm{2}}{{n}+\mathrm{1}}\int_{\mathrm{0}} ^{\:\mathrm{1}} {t}^{{n}} {ln}\left({t}\right)\right. \\ $$$$\:\:\:\:=−\mathrm{2}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}+\mathrm{1}}\left\{\left[\frac{{t}^{{n}+\mathrm{1}} }{{n}+\mathrm{1}}{ln}\left({t}\right)\right]_{\mathrm{0}} ^{\mathrm{1}} −\frac{\mathrm{1}}{{n}+\mathrm{1}}\int_{\mathrm{0}} ^{\:\mathrm{1}} {t}^{{n}} {dt}\right\} \\ $$$$\:\:\:\:=\mathrm{2}\underset{{n}=\mathrm{0}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{\left({n}+\mathrm{1}\right)^{\mathrm{3}} }=\mathrm{2}\underset{{n}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\mathrm{1}}{{n}^{\mathrm{3}} }=\mathrm{2}\zeta\left(\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:....................\:\:\:\boldsymbol{\phi}=\mathrm{2}\zeta\left(\mathrm{3}\right)\:.................... \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:..........{m}.{n}.{july}.\mathrm{1970}......... \\ $$

Question Number 133199 Answers: 4 Comments: 0