Math 7 Solution to Review Problem for Exam True or Fale? Circle ONE anwer for each Hint: For effective tudy, explain why if true and give a counterexample if fale (a) T or F : If a b and b c, then a c Let a i, b j, and c <,, > Then a b i j and b c <,, > <,, > However, a c <,, > <,, > (b) T or F: If a b, then a b a b Aume both a and b are nonzero (If either i zero the reult i obviou) If θ i the angle between a and b, then θ π ince a and b are orthogonal a b a b in θ a b in π/ a b (c) T or F: For any vector u, v in R, u v v u If θ i the angle between u and v, then u v u v in θ v u in θ v u (Or, u v v u v u v u ) (d) T or F : The vector <,, > i parallel to the plane 6x y + 4z A normal vector to the plane i n < 6,, 4 > Becaue <,, > n, the vector i parallel to n and hence perpendicular to the plane (e) T or F : If u v, then u or v For example i j but i and j (f) T or F : If u v, then u or v For example i i but i (g) T or F: If u v and u v, then u or v If both u and v are nonzero, then u v implie u and v are orthogonal But u v implie that u and v are parallel Two nonzero vector can t be both parallel and orthogonal, o at leat one of them mut be (h) T or F: The curve r(t), t, 4t i a parabola ( z ) Parametric equation for the curve are x, y t, z 4t, and ince t z we have y 4 t or 4 y z, x Thi i an equation of a parabola in the yz plane 6 (i) T or F: If κ(t) for all t, the curve i a traight line Notice that κ(t) T (t) T (t) for all t But then T(t) C, a contant vector, which i true only for a traight line (j) T or F : Different parameterization of the ame curve reult in identical tangent vector at a given point on the curve For example, r (t) < t, t > and r (t) < t, t > both repreent the ame plane curve (the line y x), but the tangent vector r (t) <, > for all t, while r (t) <, > In fact, different parametrization give parallel tangent vector at a point, but their magnitude may differ Which of the following are vector? Circle all that apply (a) [(a b)c] a A vector (b) c [(a b) c] Nonene
(c) c [(a b)c] A vector (d) (a b) c A calar Which of the following are meaningful? Circle all that apply (a) w (u v) Meaningful (b) (u v) w Nonene (c) u (v w) Meaningful 4 Find the value of x uch that the vector <,, x > and < x, 4, x > are orthogonal For the two vector to be orthogonal, we need <,, x > < x, 4, x > That i, (x) + (4) + x(x), or x or x 4 5 Let a,, b,, and u <, > (a) Show that a and b are orthogonal unit vector a b and a (b) Find the decompoition of u along a u a (u a)a, (c) Find the decompoition of u along b Similarly, u b (u b)b u b <, >,, u a +, and b +( ), Thu, u a <, >,, ( <, >, ),, Thu, 6 (a) Find an equation of the phere that pae through the point (6, -, ) and ha center (-,,) Ue the ditance formula to find the ditance between (6, -, ) and (-,,) Then, the equation for the circle i (x + ) + (y ) + (z ) 69 (b) Find the curve in which thi phere interect the yz-plane The interection of thi phere with the yz plane i the et of point on the phere whoe x coordinate i Putting x in to the equation, (y ) + (z ) 68, which repreent a circle in the yz plance with center (,, ) and radiu 68 7 For each of the following quantitie (co θ, in θ, x, y, z, and w) in the picture below, fill in the blank with the number of the expreion, taken from the lit to the right, to which it i equal Solution: co θ 5; in θ 4; x ; y ; z 7 ; w 6 8 Find an equation for the line through (4,, ) and (,, 5) The line ha direction v <,, > Letting P (4,, ), parametric equation are x 4 t, y + t, z + t 9 Find an equation for the line through (,, 4) and perpendicular to the plane x y + 5z A direction vector for the line i a normal vector for the plane, n <,, 5 >, and parametric equation for the line are x + t, y t, z 4 + 5t Find an equation of the plane through (,, ) and parallel to x + 4y z Since the two plane are parallel, they will have the ame normal vector Then we can take n <, 4, > and an equation of the plane i (x ) + 4(y ) (z )
(a) Find an equation of the plane that pae through the point A(,, ), B(,, ), and C(,, 4) The vector AB <,, 9 > and AC <,, 5 > lie in the plane, o n AB AC < 8, 4, 8 > or equivalently, <,, > i a normal vector to the plane The point A(,, ) lie on the plane o an equation for the plane i (x ) + (y ) + (z ) (b) A econd plane pae through (,, 4) and ha normal vector <, 4, > Find an equation for the line of interection of the two plane The point (,,4) lie on the econd plane, but the point alo atifie the equation of the firt plane, o the point lie on the line of interection of the plane A vector v in the direction of thi interecting line i perpendicular to the normal vector of both plane, o take v <,, > <, 4, >< 5, 5, > or jut ue <,, > Parametric equation for the line are x + t, y t, z 4 + t Find an equation of the plane through the line of interection of the plane x z and y + z and perpendicular to the plane x + y z n <,, > and n <,, > Setting z, it i eay to ee that (,, ) i a point on the line of interection of x z and y + z The direction of thi line i v n n <,, > A econd vector parallel to the deired plane i v <,, >, ince it i perpendicular to x + y z Therefore, the normal of the plane in quetion i n v v <,, > <,, > Taking (x, y, z ) (,, ), the equation we are looking for i (x ) + (y ) + z x + y + z 4 Provide a clear ketch of the following trace for the quadratic urface y plane Label your work appropriately p x + z + in the given x ; x ; y ; y ; z 4 Match the equation with their graph Give reaon for your choice (a) 8x + y + z II (b) z in x + co y I π (c) z in IV + x + y (d) z ey III 5 Decribe the et of all point P (x, y, z) atifying x + y 4 in a cylindrical coordinate In cylindrical coordinate we have x + y r, hence the inequality x + y 4 become r 4 or r and θ π That i, {(r, θ, z) : r, θ π} Thi i a olid cylinder of radiu b pherical coordinate In pherical coordinate we have x +y +z ρ and z ρ co φ Therefore, x +y ρ z ρ ρ co φ ρ in φ the inequality x +y 4 in pherical coordinate i, thu, ρ in φ 4 Notice that ince φ π, we have in φ Alo ρ, therefore ρ in φ, hence inequality ρ in φ 4 i equivalent to ρ in φ We obtain the following decription in pherical coordinate: {(ρ, θ, φ) : ρ in φ, θ π, φ π}
6 Find a vector function that repreent the curve of interection of the cylinder x + y 6 and the plane x + z 5 The projection of the curve C of interection onto the xy plane i the circle x +y 6, z So we can write x 4 co t, y 4 in t, t π From the equation of the plane, we have z 5 x 5 4 co t, o parametric equation for C are x 4 co t, y 4 in t, z 5 4 co t, t π, and the correponding vector function i r(t) < 4 co t, 4 in t, z 5 4 co t >, t π 7 Find an equation for the tangent line to the curve x in t, y in t, and z in t at the point (,, ) The curve i given by r(t) < in t, in t, in t >, o r (t) < co t, 4 co t, 6 co t > The point (,, ) correpond to t π/6, o the tangent vector there i r (π/6) <,, > Then the tangent line ha direction vector <,, > and include the point (,, ), o parametric equation are x + t, y + t, z 8 A helix circle the z axi, going from (,, ) to (,, 6π) in one turn (a) Parameterize thi helix r(t) < co t, in t, t > (Note that revolution i π, o π() 6π) (b) Calculate the length of a ingle turn For t π, r (t) 4 + 9 Thu π dt (π) (c) Find the curvature of thi helix The unit tangent vector i T(t) < in t, co t, >, o T (t) T (t) Since r (t), κ < co t, in t, > Thu, 9 (a) Sketch the curve with vector function r(t) t, co πt, in πt, t The correponding parametric equation for the curve are x t, y co πt, z in πt Since y + z, the curve i contained in a circular cylinder with axi the x axi Since x t, the curve i a helix (b) Find r (t) and r (t) Since r(t) t, co πt, in πt, r (t), π in πt, π co πt, and r (t), pi co πt, π in πt Which curve below i traced out by r(t) I Note that r() <,, > in πt, co πt, 4 t, t Find a point on the curve r(t) t +, t, 5 where the tangent line i parallel to the plane x + y 4z 5 The plane ha normal vector n <,, 4 > Since r (t) <, 4t, >, we want <, 4t, > <,, 4 > That i, + 8t, and r( 8 ) < 7 8, 6 4, 5 > Let r(t) t, (e t )/t, ln(t + ) 4
(a) Find the domain of r The expreion t, (e t )/t, and ln(t + ) are all defined when t Thu, t, t, and t + > t > Finally, the domain of r i (, ) (, ] (b) Find lim r(t) t lim r(t),, Note that in the y component we ue l Hopital Rule t (c) Find r (t) r (t) t, tet e t + t, t+ Suppoe that an object ha velocity v(t) + t, in(t), 6e t at time t, and poition r(t) <,, > at time t Find the poition, r(t), of the object at time t r(t) v(t)dt + tdt, in(t)dt, 6e t dt ( + t) / + c, co(t) + c, e t + c r() < +c, +c, +c ><,, > c, c, c So, r(t) Thu, ( + t) /, co(t) +, e t 4 If r(t) t, t co πt, in πt, evaluate r(t)dt r(t)dt t dt, t co πtdt, in πtdt, π, π 5 Find the length of the curve: x co(t), y t /, and z in(t); t ( ) ( ) ( ) dx dy dz + + dt 6 in (t) + 6 co dt dt dt (t) + 9t dt 6 + 9tdt 7 6 Reparameterize the curve r(t) < e t, e t in t, e t co t > with repect to arc length meaured from the point (,, ) in the direction of increaing t The parametric value correponding to the point (,, ) i t Since r (t) < e t, e t (co t + in t), e t (co t in t) >, r (t) e t + (co t + in t) + (co t in t) e t, and (t) t eu du (e t ) t ) ) ) ( ) ( ) ln ( + Therefore, r(t()) +, ( + in ln ( +, + co ln + 7 Find the tangent line to the curve of interection of the cylinder x + y 5 and the plane x z at the point (, 4, ) Let x(t) 5 co t, y(t) 5 in t, and z(t) 5 co t, o that r(t) < 5 co t, 5 in t, 5 co t > and r (t) < 5 in t, 5 co t, 5 in t > When (x, y, z) (, 4, ), x(t) z(t) 5 co t, and y(t) 5 in t 4, o r (t ) < 4,, 4 > i a direction vector for the line The tangent line ha parametric equation x(t) 4t, y(t) 4 + t, and z(t) 4t ANOTHER SOLUTION: Let x(t) t, y 5 x 5 t, and z(t) x(t) t Then r(t) t, (5 t) /, t In thi cae, r (t), t, 5 t When x, t, and r () <,, > So the tangent line ha parametric equation x(t) + t, y(t) 4 t, and z(t) + t 4 4 8 For the curve given by r(t) t, t, t, find (a) the unit tangent vector T(t) r (t) r (t) < t, t, > t4 + t + < t, t, > (t + ) (b) the unit normal vector T (t) < 4t, 4 t, 4t > T 4(t4 + 4t (t) + 4) (t + ) (t + ) t + and N(t) < t, t, t > t + 5
(c) the curvature κ(t) T (t) r (t) (t + ) 9 A particle move with poition function r(t) < t ln t, t, e t > Find the velocity, peed, and acceleration of the particle v(t) r (t) < + ln t,, e t > ν(t) v(t) ( + ln t) + + ( e t ) + ln t + (ln t) + e t a(t) v (t) < t,, e t > A particle tart at the origin with initial velocity <,, > and it acceleration i a(t) < 6t, t, 6t > Find it poition function 6t dt, t dt, 6t dt v(t) a(t)dt t, 4t, t + C, but <,, > v() + C, o C <,, > and v(t) t +, 4t, t + r(t) v(t)dt t + t, t 4 t, t t D But r(), o D, and r(t) t + t, t 4 t, t t Find the tangential and normal component of the acceleration vector of a particle with poition function r(t) < t, t, t > r (t) <,, t >, r (t) <,, >, r (t) + 4 + 4t 4t + 5 Then a T r (t) r (t) r (t) and a N r (t) r (t) r (t) 5 4t + 5 A flying quirrel ha poition r(t) (a) The velocity at time t, v() <,, > v(t) r (t) < + t,, t > (b) The peed at time t, ν() ν v(t) v() 4 + + 4 (c) The acceleration at time t, a() <,, > a(t) v (t) <,, > 4t 4t + 5 t + t, t, + t at time t Compute the following at time t : (d) The tangential component of acceleration at time t, a T () a T a v v, a T() <,,> <,,> +4 (e) The normal component of acceleration at time t, a N () a N a a T 5 4 OR a N a v v <,, > (f) The curvature of the quirrel path of motion at the point (,, ), κ /9 a N ν κ 9κ, κ 9 Conider the vector valued function r(t) decribing the curve hown below Put the curvature of r at A, B and C in order from mallet to larget Draw the oculating circle at thoe point B, A, C 6