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Unformatted text preview: 20C Midterm 1 Review
Martha Yip
Disclaimer: there may be typos! Vectors have a magnitude and direction.
. . . . . . k .. . . . . .j . . ............. ............. ... . .. ... ..... i basis vectors • are pairwise orthogonal (ie. i · j = i · k = j · k = 0) • (i, j, k) forms a right handed triple (ie. i × j = k) • any vector in R3 can be written as a unique linear combination of the basis vectors. (ie. v = a, b, c = ai + bj + ck) 1 Let v = a1, b1, c1, , w = a2, b2, c2 . Algebraic Add components Geometric . . .. v + w......... w ........ . . .. .... ............ ............
v Applications triangle ineq.
v + w  ≤  v  + w  . Scalar mul. λa1 , λb1 , λc1 −v . ..................... v ....................3 ....... v ...... . . ....... ........ unit vectors
ev =
1 v v  Dot prod. a1 a2 + b1 b2 + c1 c2 v · w = vw cos θ angles Cross prod. det. formula
i a1 a2 j b1 b2 k c1 c2 v × w = vw sin θ area/volume . .. .. ... w ..... .. ... .. ..
v A = v × w 
2 (v × w) ⊥ v and w .. . ............ ........... . other vector properties
magnitude:
v  = a2 + b2 + c2 direction vector from P to Q: P Q = a2 − a1, b2 − b1, c2 − c1
....Q = (a2, b2, c2) . . ..... .......... ........... • .... .... • P = (a1 , b1 , c1 ) orthogonality: parallel: v⊥w v // w ⇔ ⇔ v·w =0 v×w =0 3 vector projection/decomposition . .. . .. u . .. . . .. . . . . .. . . .. . . . .. . . . proj= . .. . . . . ... . . ... . ... has length .. . ..... ... . .v = jj cos 4 vector projection/decomposition . .. .. . .. ..u .. .. .. . ... .. . . .. .. . . .. .. . .. . . .. . . ... .... . .. . . u·v . ... v projv u = . . . v 2 ... . . ..... . u·v . θ.. . . ..... has length = u cos θ .. .... v  .... . .v 4a vector projection/decomposition .. .. . .. .. . .. ..u .. .. .. . ... .. .. . . .. .. . .. . .. .. . .. . .. . .. . . .. ... .... . .. .. . . . ... .. . . u·v . .. ... v projv u = . ... . .. .. v 2 . .... . ..... .... . θ.. .. . .. u·v .... ..... ... .. . has length = u cos θ .. v v⊥ . v  4b vector projection/decomposition . .u ... .. ...... . .. .. . .. . . . .. .. .. . .. .. .. .. .. .. .. . .. . .. .. . .. .. . . . .. ... .... . .. . .. .. . . . . . ... .... ... . . u·v u · v⊥ ... ... ⊥ . ... v projv u = projv u = v ... . ... ... . .. v 2  v ⊥ 2 ... . θ . ... . . ... . .... .. .. ... . .. .... ... u·v ...... .... v has length = u cos θ has length u sin θ v⊥ . v  .. ..
⊥ 4c vector projection/decomposition . .u ... .. ...... . .. .. . .. . . . .. .. .. . .. .. .. .. .. .. .. . .. . .. .. . .. .. . . . .. ... .... u// . .. . .. .. . . . u⊥ ..... ... ... .. . . u·v u · v⊥ .. . .. ⊥ . ... v projv u = projv u = v ... . ... ... . .. v 2  v ⊥ 2 ... . θ . ... . . ... . .... .. .. ... . .. .... ... u·v ...... .... v has length = u cos θ has length u sin θ v⊥ . v  .. ..
⊥ u = u// + u⊥ 4d Vectors check your understanding 1. The distance between the points (−1, 3, 5) and (x, y, z ) is . 2. True or False? The vectors v and −2v are parallel. 3. True or False? The vectors v and −2v point in the same direction. 4. If v × w = 2, 2, 1 , then w × v = . 5. 4, 8, 2 × 2, 4, 1 = . 5 Vectors check your understanding 1. The distance between the points (−1, 3, 5) and (x, y, z ) is (x + 1)2 + (y − 3)2 + (z − 5)2. 2. True or False? The vectors v and −2v are parallel. 3. True or False? The vectors v and −2v point in the same direction. 4. If v × w = 2, 2, 1 , then w × v = . 5. 4, 8, 2 × 2, 4, 1 = . 5a Vectors check your understanding 1. The distance between the points (−1, 3, 5) and (x, y, z ) is (x + 1)2 + (y − 3)2 + (z − 5)2. 2. True or False? The vectors v and −2v are parallel. 3. True or False? The vectors v and −2v point in the same direction. 4. If v × w = 2, 2, 1 , then w × v = . 5. 4, 8, 2 × 2, 4, 1 = . 5b Vectors check your understanding 1. The distance between the points (−1, 3, 5) and (x, y, z ) is (x + 1)2 + (y − 3)2 + (z − 5)2. 2. True or False? The vectors v and −2v are parallel. 3. True or False? The vectors v and −2v point in the same direction. 4. If v × w = 2, 2, 1 , then w × v = . 5. 4, 8, 2 × 2, 4, 1 = . 5c Vectors check your understanding 1. The distance between the points (−1, 3, 5) and (x, y, z ) is (x + 1)2 + (y − 3)2 + (z − 5)2. 2. True or False? The vectors v and −2v are parallel. 3. True or False? The vectors v and −2v point in the same direction. 4. If v × w = 2, 2, 1 , then w × v = − 2, 2, 1 . 5. 4, 8, 2 × 2, 4, 1 = . 5d Vectors check your understanding 1. The distance between the points (−1, 3, 5) and (x, y, z ) is (x + 1)2 + (y − 3)2 + (z − 5)2. 2. True or False? The vectors v and −2v are parallel. 3. True or False? The vectors v and −2v point in the same direction. 4. If v × w = 2, 2, 1 , then w × v = − 2, 2, 1 . 5. 4, 8, 2 × 2, 4, 1 = 0. 5e 1. Do these make sense? u · (v × w ), (a · b) × c, (a × b) + k, (u × v) × w 2. What is projv v? 3. What is the diﬀerence between projv u and projev u? 4. If θ is the angle between v and w, which of the following is equal to cos θ? u · v, u · ev , eu · ev . 6 1. Do these make sense? u · (v × w ), Y (a · b) × c, N (a × b) + k, N (u × v ) × w Y 2. What is projv v? v 3. What is the diﬀerence between projv u and projev u? They are the same. 4. If θ is the angle between v and w, which of the following is equal to cos θ? u · v, u · ev , eu · ev . 6a Planes are determined by a point P0 and a normal vector n. .. . .. . . .. . . .. . . .. . . .. . . . .. . . . . . . • . P = (x, y, z ) . . . . . . . . . . . . . . . . . . . . . . . . . . . n= . . . . . .................................. a, b, c . . •................................. . . ... . . . . . ... . . P0 = (x0, y0, z0) . .. .. .. .. .. .. .. .. .. . . . . n · P P0 = 0 a(x − x0) + b(y − y0) + c(z − z0) = 0 7 Planes check your understanding 1. 3(x − 2) − 2(y − 1) + 4z = 0 is the equation of a plane through the point with the vector perpendicular to the plane. 2. True or False? 3 noncollinear points uniquely determine a plane. 3. If the noncollinear points A, B , and C lie on a plane, then a vector . normal to the plane is 4. The angle between two intersecting planes with normal vectors n1 and n2 is . 5. The direction of the line of intersection of two planes with normal vectors n1 and n2 is .
8 Planes check your understanding 1. 3(x − 2) − 2(y − 1) + 4z = 0 is the equation of a plane through the point (2, 1, 0) with the vector 3, −2, 4 p erpendicular to the plane. 2. True or False? 3 noncollinear points uniquely determine a plane. 3. If the noncollinear points A, B , and C lie on a plane, then a vector . normal to the plane is 4. The angle between two intersecting planes with normal vectors n1 and n2 is . 5. The direction of the line of intersection of two planes with normal vectors n1 and n2 is .
8a Planes check your understanding 1. 3(x − 2) − 2(y − 1) + 4z = 0 is the equation of a plane through the point (2, 1, 0) with the vector 3, −2, 4 p erpendicular to the plane. 2. True or False? 3 noncollinear points uniquely determine a plane. 3. If the noncollinear points A, B , and C lie on a plane, then a vector . normal to the plane is 4. The angle between two intersecting planes with normal vectors n1 and n2 is . 5. The direction of the line of intersection of two planes with normal vectors n1 and n2 is .
8b Planes check your understanding 1. 3(x − 2) − 2(y − 1) + 4z = 0 is the equation of a plane through the point (2, 1, 0) with the vector 3, −2, 4 p erpendicular to the plane. 2. True or False? 3 noncollinear points uniquely determine a plane. 3. If the noncollinear points A, B , and C lie on a plane, then a vector normal to the plane is AB × AC 4. The angle between two intersecting planes with normal vectors n1 . and n2 is 5. The direction of the line of intersection of two planes with normal vectors n1 and n2 is .
8c Planes check your understanding 1. 3(x − 2) − 2(y − 1) + 4z = 0 is the equation of a plane through the point (2, 1, 0) with the vector 3, −2, 4 p erpendicular to the plane. 2. True or False? 3 noncollinear points uniquely determine a plane. 3. If the noncollinear points A, B , and C lie on a plane, then a vector normal to the plane is AB × AC 4. The angle between two intersecting planes with normal vectors n1 and n2 is the angle between n1 and n2. 5. The direction of the line of intersection of two planes with normal . vectors n1 and n2 is
8d Planes check your understanding 1. 3(x − 2) − 2(y − 1) + 4z = 0 is the equation of a plane through the point (2, 1, 0) with the vector 3, −2, 4 p erpendicular to the plane. 2. True or False? 3 noncollinear points uniquely determine a plane. 3. If the noncollinear points A, B , and C lie on a plane, then a vector normal to the plane is AB × AC 4. The angle between two intersecting planes with normal vectors n1 and n2 is the angle between n1 and n2. 5. The direction of the line of intersection of two planes with normal vectors n1 and n2 is n1 × n2 .
8e 1. Which of the following planes are NOT parallel to x + y + z = 1? 2x + 2y + 2z = 1, x + y + z = 3, x − y + z = 0. 2. To which coordinate plane is the plane y = 1 parallel? 3. Supppose a plane P has normal n, and a line has direction v, and they both pass through 0. Also, n · v = 0. Which is true: is contained in P is orthogonal to P 9 1. Which of the following planes are NOT parallel to x + y + z = 1? 2x + 2y + 2z = 1, x + y + z = 3, x − y + z = 0. 2. To which coordinate plane is the plane y = 1 parallel? xzplane 3. Supppose a plane P has normal n, and a line has direction v, and they both pass through 0. Also, n · v = 0. Which is true: is contained in P is orthogonal to P 9a Vector parametrizations r(t) = x(t), y (t), z (t) In twodimensions, r(t) = x(t), y (t) , eliminate t to get y = f (x). Parametrizations of common curves
Lines: Ellipses/Circles: Cylindrical helix: Conical helix: r(t) = x0 + at, y0 + bt, z0 + ct r(t) = x0 + a cos(t), y0 + b sin(t), z0 r(t) = a cos(t), a sin(t), t r(t) = t cos(t), t sin(t), t 10 Diﬀerent parametrization, diﬀerent motion
for t ∈ (−∞, ∞): Consider r1(t) = t3, t6 , r2(t) = t2, t4 , r3(t) = cos t, cos2 t . The underlying curve in each case is y = x2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. . . . .. .. . . .. . . . . .. . .. .. . .. . ... . ... ........................ . . ................. ..................... ................. . . . r1(t): x = t3 can be any real number. It travels along the whole parabola, coming from the left to the right. (t): note x = t2 ! 0. It travels along the positive part of the parabola, coming from the right, hits the origin at t = 0, then goes out to the right again. (t) note 1 cos t 1. It bounces back and forth on the nite segment of the parabola from x = 1 to x = 1.
11 Diﬀerent parametrization, diﬀerent motion
for t ∈ (−∞, ∞): Consider r1(t) = t3, t6 , r2(t) = t2, t4 , r3(t) = cos t, cos2 t . The underlying curve in each case is y = x2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. . . . .. .. . . .. . . . . .. . .. .. . .. . ... . ... ........................ . . ................. ..................... ................. . . . r1(t): x = t3 can be any real number. It travels along the whole parabola, coming from the left to the right. r2(t): note x = t2 ≥ 0. It travels along the positive part of the parabola, coming from the right, hits the origin at t = 0, then goes out to the right again. (t) note 1 cos t 1. It bounces back and forth on the nite segment of the parabola from x = 1 to x = 1.
11a Diﬀerent parametrization, diﬀerent motion
for t ∈ (−∞, ∞): Consider r1(t) = t3, t6 , r2(t) = t2, t4 , r3(t) = cos t, cos2 t . The underlying curve in each case is y = x2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . .. . . . .. .. . . .. . . . . .. . .. .. . .. . ... . ... ........................ . . ................. ..................... ................. . . . r1(t): x = t3 can be any real number. It travels along the whole parabola, coming from the left to the right. r2(t): note x = t2 ≥ 0. It travels along the positive part of the parabola, coming from the right, hits the origin at t = 0, then goes out to the right again. r3(t) note −1 ≤ cos t ≤ 1. It bounces back and forth on the ﬁnite segment of the parabola from x = −1 to x = 1.
11b Arclength formula comes from approximation by line segments If r(t) = x(t), y (t), z (t) , then the length of the curve between r(a) and r(b) is
b s= a b dx 2 dy 2 dz 2 r (t)dt. + + dt = dt dt dt a In the special case where z = 0, x = t, so y = f (x) = f (t), then
b s= a 1 + f (x)2dx. 12 Parametrizations check your understanding 1. The underlying curve of r(t) = cos(−2t), sin(−2t) is a . direction. It completes one full cycle in It travels in the seconds. 2. What is the center of the circle with parametrization r(t) = −2 + cos(t), 2, 3 − sin(t) .
3. If r(t) = x(t), y (t) , match the following: (i) dx , dt (a) slope of tangent line (b) vertical rate of change wrt time (c) horizontal rate of change wrt time (ii) dy , dt
dy (iii) dx . 4. If r(t) = x(t), y (t) , the slope of the tangent line at r(t) is
13 . Parametrizations check your understanding 1. The underlying curve of r(t) = cos(−2t), sin(−2t) is a circle. It travels in the clockwise direction. It completes one full cycle in π seconds. 2. What is the center of the circle with parametrization r(t) = −2 + cos(t), 2, 3 − sin(t) .
(−2, 2, 3) (and it’s in the y = 2 plane) 3. If r(t) = x(t), y (t) , match the following: (i) dx , dt (a) slope of tangent line (iii) (b) vertical rate of change wrt time (ii) (c) horizontal rate of change wrt time (i) 4. If r(t) = x(t), y (t) , the slope of the tangent line at r(t) is dx/dt .
13a (ii) dy , dt dy (iii) dx . dy/dt Motion in nspace
position vector velocity vector acceleration vector displacement between t = a, t = b distance traveled (since time = a) speed tangential acceleration r(t) v(t) = r (t) a(t) = r (t) r(b) − r(a)
t s( t) = a v(u) du v (t) = v(t) = s (t) a(t) = s (t) a(t) = s (t) T(t) + κ(s (t))2 N(t)
tangential component normal component 14 tangential acceleration .. .. a .. ...... .. ....... . . .. .. . .. . . . .. .. . . .. .. .. .. .. .. . .. . .. .. . .. .. .. . .. . ... .... s T . .. . . .. . . .. . . . . .... ... .. ... . . . ... . κ(s )2 N .... ... . ... ... . ..... . .... . .... .... ... .. . .. .... ... . ... .. .. • N .. T a = s T + κ(s )2 N
s(t) is the speed, T(t) is the direction of travel at time t. 15 Motion in nspace, cont’d
In linear motion described by the position function s(t), the particle is speeding up if v (t) and a(t) are both positive or both negative (have same direction), if v (t) and a(t) have opposite signs. slowing down For motion in higher dimensions, s T is the component of a in the tangential direction. The particle is speeding up slowing down if v and s T have same direction, (ie. v · a > 0) if v and s T have opposite direction (ie. v · a < 0). 16 Calculus of vectorvalued functions do in components A parametrization of the tangent line to the curve at r(t0) is L(t) = r(t0) + tv(t0) as long as v(t0) = 0. Fundamental Theorem of Calculus for vector valued functions
b a r(t)dt = R(b) − R(a) if R (t) = r(t). 17 Initial value problems Given a(t), with initial velocity v0, and initial position r0, ﬁnd v(t) and r(t). v(t) = r(t) = t 0 t 0 a(u)du + v0, v(u)du + r0. 18 Motion in nspace check your understanding 1. True or False? The derivative of the cross product, is the cross product of the derivative. 2. True or False? The derivative of a vectorvalued function is the slope of the tangent line, just as in the scalar case. 3. A particle travels along y = x2 with constant speed 1 cm/s. What is the distance traveled during the ﬁrst minute? 4. The velocity vector r (t) is to the curve at r(t). 5. True or False: If a particle travels with constant speed, its acceleration vector necessarily zero.
19 Motion in nspace check your understanding 1. True or False? The derivative of the cross product, is the cross product of the derivative. product rule 2. True or False? The derivative of a vectorvalued function is the slope of the tangent line, just as in the scalar case. derivative of a vector function is not a scalar 3. A particle travels along y = x2 with constant speed 1 cm/s. What is the distance traveled during the ﬁrst minute? 60cm. does not depend on the path 4. The velocity vector r (t) is tangent to the curve at r(t). 5. True or False: If a particle travels with constant speed, its acceleration vector necessarily zero. direction of a may change
19a 1. For a particle traveling in uniform circular motion, its vector always points towards the center of the circle. 2. True or False: If the speed is constant, then the acceleration and velocity vectors are orthogonal. 20 1. For a particle traveling in uniform circular motion, its acceleration vector always points towards the center of the circle. 2. True or False: If the speed is constant, then the acceleration and velocity vectors are orthogonal. particle is momentarily not speeding up or slowing down 20a Study hard and Good luck! 21 ...
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This note was uploaded on 03/28/2011 for the course MATH 20C taught by Professor Helton during the Winter '08 term at UCSD.
 Winter '08
 Helton
 Vectors

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