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Vector CalculusQuestion and Answers: Page 4

Question Number 102474 Answers: 1 Comments: 0

Un=(1+(√2))^n show that we have p_n ∈N / U_n =(√p_n )+(√(p_n +1))

$$\boldsymbol{{U}}{n}=\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)^{{n}} \\ $$$${show}\:{that}\:{we}\:{have}\:\boldsymbol{{p}}_{\boldsymbol{{n}}} \in\mathbb{N}\:/ \\ $$$$\boldsymbol{{U}}_{\boldsymbol{{n}}} =\sqrt{{p}_{{n}} }+\sqrt{{p}_{{n}} +\mathrm{1}} \\ $$

Question Number 100320 Answers: 1 Comments: 0

Evaluate ∫∫_s F^→ .n^ dS where F^→ =4xi^ −2y^2 j^ +z^2 k^ and S is the surface of the cylinder bounded by x^2 +y^2 =4 ,z = 0 and z=3 .

$$\mathrm{Evaluate}\:\int\underset{\mathrm{s}} {\int}\:\overset{\rightarrow} {\mathrm{F}}.\hat {\mathrm{n}}\:\mathrm{dS}\:\mathrm{where}\:\overset{\rightarrow} {\mathrm{F}}=\mathrm{4x}\hat {\mathrm{i}}\:−\mathrm{2y}^{\mathrm{2}} \hat {\mathrm{j}}\:+\mathrm{z}^{\mathrm{2}} \hat {\mathrm{k}}\: \\ $$$$\mathrm{and}\:\mathrm{S}\:\mathrm{is}\:\mathrm{the}\:\mathrm{surface}\:\mathrm{of}\:\mathrm{the}\:\mathrm{cylinder} \\ $$$$\mathrm{bounded}\:\mathrm{by}\:\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} =\mathrm{4}\:,\mathrm{z}\:=\:\mathrm{0}\:\mathrm{and}\:\mathrm{z}=\mathrm{3}\:. \\ $$

Question Number 100074 Answers: 1 Comments: 0

Question Number 98119 Answers: 0 Comments: 1

Find the shortest distance between the skew lines ((x−3)/3) = ((8−y)/1) = ((z−3)/1) and ((x+3)/(−3)) = ((y+7)/2) = ((z−6)/4) .

$$\mathrm{Find}\:\mathrm{the}\:\mathrm{shortest}\:\mathrm{distance}\:\mathrm{between}\:\mathrm{the} \\ $$$$\mathrm{skew}\:\mathrm{lines}\:\frac{\mathrm{x}−\mathrm{3}}{\mathrm{3}}\:=\:\frac{\mathrm{8}−\mathrm{y}}{\mathrm{1}}\:=\:\frac{\mathrm{z}−\mathrm{3}}{\mathrm{1}}\:\mathrm{and}\: \\ $$$$\frac{\mathrm{x}+\mathrm{3}}{−\mathrm{3}}\:=\:\frac{\mathrm{y}+\mathrm{7}}{\mathrm{2}}\:=\:\frac{\mathrm{z}−\mathrm{6}}{\mathrm{4}}\:. \\ $$

Question Number 97901 Answers: 2 Comments: 2

Question Number 97353 Answers: 5 Comments: 0

Question Number 95509 Answers: 2 Comments: 0

find the angle of plane 2x−y+2z=1 and x+3y−2z = 2

$$\mathrm{find}\:\mathrm{the}\:\mathrm{angle}\:\mathrm{of}\:\mathrm{plane} \\ $$$$\mathrm{2x}−\mathrm{y}+\mathrm{2z}=\mathrm{1}\:\mathrm{and}\:\mathrm{x}+\mathrm{3y}−\mathrm{2z}\:=\:\mathrm{2} \\ $$

Question Number 92143 Answers: 0 Comments: 1

fond∫((2x)/(1+x^2 ))dx

$${fond}\int\frac{\mathrm{2}{x}}{\mathrm{1}+{x}^{\mathrm{2}} }{dx} \\ $$

Question Number 92138 Answers: 2 Comments: 0

∫(dx/(1+x^2 ))

$$\int\frac{{dx}}{\mathrm{1}+{x}^{\mathrm{2}} } \\ $$

Question Number 84969 Answers: 0 Comments: 0

let c is a constant vector and r^→ =xi^ +yj^ +zk^ then proved that grad ∣c×r^→ ∣^n =n∣c×r^→ ∣^(n−2) c×(r^→ ×c).

$$\mathrm{let}\:\mathrm{c}\:\mathrm{is}\:\mathrm{a}\:\mathrm{constant}\:\mathrm{vector}\:\mathrm{and}\:\overset{\rightarrow} {\mathrm{r}}=\mathrm{x}\hat {\mathrm{i}}+\mathrm{y}\hat {\mathrm{j}}+\mathrm{z}\hat {\mathrm{k}}\:\mathrm{then}\:\mathrm{proved}\:\mathrm{that}\:\mathrm{grad}\:\mid\mathrm{c}×\overset{\rightarrow} {\mathrm{r}}\mid^{\mathrm{n}} =\mathrm{n}\mid\mathrm{c}×\overset{\rightarrow} {\mathrm{r}}\mid^{\mathrm{n}−\mathrm{2}} \mathrm{c}×\left(\overset{\rightarrow} {\mathrm{r}}×\mathrm{c}\right). \\ $$

Question Number 84624 Answers: 1 Comments: 0

find grad r^m where r=x^2 +y^2 +z^2

$$\mathrm{find}\:\mathrm{grad}\:\mathrm{r}^{\mathrm{m}} \:\:\mathrm{where}\:\mathrm{r}=\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} \\ $$

Question Number 80402 Answers: 0 Comments: 0

Question Number 78522 Answers: 0 Comments: 10

what is the line passing through (2,2,1) and parallel to 2i^ − j^ − k^ ?

$${what}\:{is}\:{the}\: \\ $$$${line}\:{passing}\:{through}\:\left(\mathrm{2},\mathrm{2},\mathrm{1}\right) \\ $$$${and}\:{parallel}\:{to}\:\mathrm{2}\hat {{i}}\:−\:\hat {{j}}\:−\:\hat {{k}}\:? \\ $$

Question Number 76302 Answers: 1 Comments: 0

given vektor a=(3,x,−2) b=(−6,−2,y) . what the value x and y if a and b are parallel?

$${given}\:{vektor}\:{a}=\left(\mathrm{3},{x},−\mathrm{2}\right) \\ $$$${b}=\left(−\mathrm{6},−\mathrm{2},{y}\right)\:.\:{what}\:{the}\:{value}\:{x}\: \\ $$$${and}\:{y}\:{if}\:{a}\:{and}\:{b}\:{are}\:{parallel}? \\ $$

Question Number 75101 Answers: 4 Comments: 2

the vector equations of two lines L_1 and L_2 is given by L_1 :r= i−j+3k + λ(i−j +k) L_2 : r= 2i+aj + 6k + μ(2i + j + 3k) where a,λ,μ are real constants. given that L_1 and L_2 intersect find a. the value of the constant a. b. the position vector of the point of intersection between L_1 and L_2 c. the cosine of the acute angle between L_1 and L_2 please help

$${the}\:{vector}\:{equations}\:{of}\:{two}\:{lines}\:{L}_{\mathrm{1}} \:{and}\:{L}_{\mathrm{2}} \:{is}\:{given}\:{by} \\ $$$$\:{L}_{\mathrm{1}} :{r}=\:\boldsymbol{{i}}−\boldsymbol{{j}}+\mathrm{3}\boldsymbol{{k}}\:+\:\lambda\left(\boldsymbol{{i}}−\boldsymbol{{j}}\:+\boldsymbol{{k}}\right) \\ $$$${L}_{\mathrm{2}} \::\:{r}=\:\mathrm{2}\boldsymbol{{i}}+{a}\boldsymbol{{j}}\:+\:\mathrm{6}\boldsymbol{{k}}\:+\:\mu\left(\mathrm{2}\boldsymbol{{i}}\:+\:\boldsymbol{{j}}\:+\:\mathrm{3}\boldsymbol{{k}}\right) \\ $$$${where}\:{a},\lambda,\mu\:{are}\:{real}\:{constants}. \\ $$$${given}\:{that}\:{L}_{\mathrm{1}} \:{and}\:{L}_{\mathrm{2}} \:{intersect}\:{find} \\ $$$${a}.\:\:{the}\:{value}\:{of}\:{the}\:{constant}\:{a}. \\ $$$${b}.\:\:{the}\:{position}\:{vector}\:{of}\:{the}\:{point}\:{of}\: \\ $$$${intersection}\:{between}\:{L}_{\mathrm{1}} \:{and}\:{L}_{\mathrm{2}} \\ $$$${c}.\:{the}\:{cosine}\:{of}\:{the}\:{acute}\:{angle}\:{between}\:{L}_{\mathrm{1}} \:{and}\:{L}_{\mathrm{2}} \\ $$$${please}\:{help} \\ $$$$ \\ $$

Question Number 71198 Answers: 0 Comments: 0

A particle P is projected from a point O at the edge of a cliff 60m from the sea with a velocity of 30ms^(−1) . When P is at a point B where OB is a horizontal, another particle Qsuch that P and Q hit the sea simultaneously at thesame point A. Gven that they strike the sea 6seconds after P was fired ^ calculate a) the sine of the angle of elevation of projection. b) the distance from A to O. c) the time of flight of Q. d) the Range . (take g = 10ms^(−2) ) please help

$${A}\:{particle}\:{P}\:{is}\:{projected}\:{from}\:\:{a}\:{point}\:{O}\:{at}\:\:{the}\:{edge}\:{of}\:{a}\:{cliff}\:\mathrm{60}{m} \\ $$$${from}\:{the}\:{sea}\:{with}\:{a}\:{velocity}\:{of}\:\mathrm{30}{ms}^{−\mathrm{1}} .\:{When}\:{P}\:{is}\:{at}\:{a}\:{point}\:{B} \\ $$$${where}\:{OB}\:{is}\:{a}\:{horizontal},\:{another}\:{particle}\:{Qsuch}\:{that}\: \\ $$$${P}\:{and}\:{Q}\:{hit}\:{the}\:{sea}\:{simultaneously}\:{at}\:{thesame}\:{point}\:{A}.\:{Gven}\:{that}\:{they} \\ $$$${strike}\:{the}\:{sea}\:\mathrm{6}{seconds}\:{after}\:{P}\:{was}\:{fired}\bar {\:}\:{calculate} \\ $$$$\left.{a}\right)\:{the}\:{sine}\:{of}\:{the}\:{angle}\:{of}\:{elevation}\:{of}\:{projection}. \\ $$$$\left.{b}\right)\:{the}\:{distance}\:{from}\:{A}\:{to}\:{O}. \\ $$$$\left.{c}\right)\:{the}\:{time}\:{of}\:{flight}\:{of}\:{Q}. \\ $$$$\left.{d}\right)\:{the}\:{Range}\:. \\ $$$$\left({take}\:{g}\:=\:\mathrm{10}{ms}^{−\mathrm{2}} \right)\: \\ $$$${please}\:{help}\: \\ $$

Question Number 70876 Answers: 0 Comments: 0

Prove that The necessary and sufficient condition that the curve be plane (curve) is [r′,r′′,r′′′]=0. OR A curve is plane curve iff τ=0.

$$\mathrm{Prove}\:\mathrm{that} \\ $$$$\mathrm{The}\:\mathrm{necessary}\:\mathrm{and}\:\mathrm{sufficient}\:\mathrm{condition} \\ $$$$\mathrm{that}\:\mathrm{the}\:\mathrm{curve}\:\mathrm{be}\:\mathrm{plane}\:\left(\mathrm{curve}\right)\:\mathrm{is} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\left[\boldsymbol{\mathrm{r}}',\boldsymbol{\mathrm{r}}'',\boldsymbol{\mathrm{r}}'''\right]=\mathrm{0}. \\ $$$$\:\mathrm{OR}\:\:\:\: \\ $$$$\mathrm{A}\:\mathrm{curve}\:\mathrm{is}\:\mathrm{plane}\:\mathrm{curve}\:\mathrm{iff}\:\tau=\mathrm{0}. \\ $$

Question Number 68836 Answers: 1 Comments: 0

Show that the graph of r = (sin t)i + (2cos t)j + ((√3)sin t)k is a circle

$$\mathrm{Show}\:\mathrm{that}\:\mathrm{the}\:\mathrm{graph}\:\mathrm{of} \\ $$$$\boldsymbol{{r}}\:=\:\left(\mathrm{sin}\:{t}\right)\boldsymbol{{i}}\:+\:\left(\mathrm{2cos}\:{t}\right)\boldsymbol{{j}}\:+\:\left(\sqrt{\mathrm{3}}\mathrm{sin}\:{t}\right)\boldsymbol{{k}} \\ $$$$\mathrm{is}\:\mathrm{a}\:\mathrm{circle} \\ $$

Question Number 65797 Answers: 1 Comments: 0

find the constant a,b and c so that the direction derivative of Φ=axy^2 +byz+cz^2 x^3 at (1,2,−1) has a maximum of magnitude 64 jn a direction parallel to the z axis.

$${find}\:{the}\:{constant}\:\:{a},{b}\:{and}\:\:{c}\:\:{so} \\ $$$${that}\:{the}\:{direction}\:{derivative}\:{of} \\ $$$$\Phi={axy}^{\mathrm{2}} +{byz}+{cz}^{\mathrm{2}} {x}^{\mathrm{3}} \:{at}\:\left(\mathrm{1},\mathrm{2},−\mathrm{1}\right) \\ $$$${has}\:{a}\:{maximum}\:{of}\:{magnitude} \\ $$$$\mathrm{64}\:{jn}\:{a}\:{direction}\:{parallel}\:{to}\:{the} \\ $$$${z}\:{axis}. \\ $$

Question Number 65261 Answers: 0 Comments: 1

solve4−xy+yz−xz

$${solve}\mathrm{4}−{xy}+{yz}−{xz} \\ $$

Question Number 64456 Answers: 0 Comments: 1

Question Number 64011 Answers: 1 Comments: 3

lim_(x→0) ((x^x −1)/(xlnx))

$$\mathrm{li}\underset{\mathrm{x}\rightarrow\mathrm{0}} {\mathrm{m}}\frac{\mathrm{x}^{\mathrm{x}} −\mathrm{1}}{\mathrm{xlnx}} \\ $$

Question Number 62517 Answers: 4 Comments: 2

Question Number 61258 Answers: 1 Comments: 2

Calculate, using cartesian coodinates, the following integrals: 1) ∫∫_D dxdy being D={ (x,y)∈R^2 /0≤x≤(1/2),y+x≤1,y≥0} 2) ∫∫_D x^3 ydxdy being D={(x,y)∈R^2 /0≤x≤(1/2),y+x≤1,y≥0} 3) ∫∫_D (x/y)dxdy being D={(x,y)∈R^2 /xy≤16,x≥y,x−6≤y,x≥0,y≥1} Help please!

$$\boldsymbol{{C}}{alculate},\:{using}\:{cartesian}\:{coodinates},\:{the}\:{following} \\ $$$${integrals}: \\ $$$$ \\ $$$$\left.\mathrm{1}\right)\:\int\int_{{D}} {dxdy}\:\:{being}\:\:{D}=\left\{\:\left({x},{y}\right)\in{R}^{\mathrm{2}} /\mathrm{0}\leqslant{x}\leqslant\frac{\mathrm{1}}{\mathrm{2}},{y}+{x}\leqslant\mathrm{1},{y}\geqslant\mathrm{0}\right\} \\ $$$$\left.\mathrm{2}\right)\:\int\int_{{D}} {x}^{\mathrm{3}} {ydxdy}\:\:{being}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\mathrm{0}\leqslant{x}\leqslant\frac{\mathrm{1}}{\mathrm{2}},{y}+{x}\leqslant\mathrm{1},{y}\geqslant\mathrm{0}\right\} \\ $$$$\left.\mathrm{3}\right)\:\int\int_{{D}} \frac{{x}}{{y}}{dxdy}\:\:{being}\:{D}=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /{xy}\leqslant\mathrm{16},{x}\geqslant{y},{x}−\mathrm{6}\leqslant{y},{x}\geqslant\mathrm{0},{y}\geqslant\mathrm{1}\right\} \\ $$$$ \\ $$$${Help}\:\:{please}! \\ $$

Question Number 60015 Answers: 0 Comments: 0

Question Number 60013 Answers: 0 Comments: 0

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