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Ch. 10 Review
❖ Be able to visualize 3D coordinate systems.
❖ If k is a constant, x=k is a place parallel to the yz-plane, y=k is a plane parallel to the xz-plane, and z=k is a plane parallel to the xy-plane.
❖ Distance between point P1(x1,y1,z1,) and P2(x2,y2,z2) If you want to learn more check out Which economic system refers to productive resources publicly owned and allocated?
⮚ sqrt((x2-x1)^2 + (yz-y1)^2 + (z2-z1)^2)
We also discuss several other topics like What is the overall effect of placebo in a person?
❖ Equation of a sphere with center C(h,k,l) and radius r is
⮚ (x-h)^2 + (y-k)^2 + (z-l)^2 = r^2
⮚ If the center is the origin ‘O’, then the equation is x^2 + y^2 + z^2 = r^2
❖ Collinear: Sum of two smaller distances equals the third (when dealing with three points).
Ex. Problem: Show that x^2 + y^2 + z^2 + 4x – 6y + 2z + 6=0 is the equation of a sphere, and find its center and radius. (pg. 540) Don't forget about the age old question of What correlation is represented in ceteris paribus?
❖ Triangle Law: AC=AB+BC
❖ Vector u – vector v= u + (-v)
❖ Given points A(x1, y1, z1) and B(x2, y2, z2), then vector a between them is a=(x2-x1,y2-y1,z2-z1) ❖ Magnitude/Length of a=<a1,a2,a3> is |a|=sqrt(a12 + a22 + a32)
❖ Know how to add/subtract vectors, as well as multiply a vector by a scalar If you want to learn more check out Whata re the properties of valence electrons?
❖ Know the vector properties (pg. 546)
❖ i, j, k notation: i=<1,0,0> j=<0,1,0> k=<0,0,1>
Ex. Problem: Find the unit vector in the direction of the vector 2i-j-2k. (pg. 548)
❖ Dot Product: a dot b = a1b1 + a2b2 + a3b3
❖ Know the properties of the dot product (pg. 551)
❖ If theta is the angle between the vectors a and b, then a dot b = |a||b|cosƟ ⮚ cosƟ = (a dot b)/|a||b|
❖ Two vectors are orthogonal if their dot product=0 If you want to learn more check out What does condensation do to water?
❖ Projections (Pg. 554)
⮚ Scalar projection of b onto a: compab=(a dot b)/|a|
⮚ Vector projection of b onto a: projab=((a dot b)/|a|)(a/|a|) = ((a dot b)/|a|2)*aIf you want to learn more check out What comprises a community?
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Ex. Problem: A wagon is pulled a distance of 100m along a horizontal path by a constant force of 70N. The handle of the wagon is held at an angle of 35° above the horizontal. Find the work done by the force. (pg. 555).
❖ Cross Product: a X b = <a2b3 – a3b2, a3b1 – a1b3,a1b2-a2b3>
⮚ Don’t forget the j component is negative
❖ Vector a X b is orthogonal to both a and b.
❖ If Ɵ is the angle between a and b (so 0<Ɵ<pi), then |a X b| = |a||b|sinƟ
❖ Two nonzero vectors a and b are parallel if and only if a X b = 0
⮚ The length of the cross product a X b is equal to the area of the parallelogram determined by a and b.
❖ Know the cross product properties (pg. 562)
❖ Volume of parallelepiped is V = |a dot (b X c)|(also called scalar triple product) ❖ Torque = r X F where r is the radius from the applied force F
⮚ |t| = |r X F| = |r||F|sinƟ
Ex. Problem: Use the scalar triple product to show that the vectors a = <1,4,-7>, b=<2,-1,4>, and c=<0,- 9,18> are coplanar (pg. 563).
❖ A line L is determined when we know a point P0(x0,y0,z0) on L and the direction of L. ❖ r = r0 + tv
❖ Two vectors are equal if and only if corresponding components are equal (parametric equations) ⮚ X = xo + at y=y0 + bt z=z0 + ct
❖ Solve for t to get the symmetric equations
⮚ (x-x0)/a = (y-y0)/b = (z-z0)/c
❖ A line segment from r0 to r1 is given by the vector equation:
⮚ r(t) = (1-t)r0 + tr1
❖ Vector equation of the plane:
⮚ N dot (r – r0) = 0 or n dot r = n dot r0 where r and r0 are vectors
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❖ When n = <a,b,c> and r = <x,y,z> and r0 = <x0,y0,z0> then the vector equation of the plane is: ⮚ a(x-x0) + b(y-y0) + c(z-z0) = 0
❖ The linear equation is ax + by + cz + d = 0 where d = -(ax0 + by0 + cz0).
❖ Distance from a point to a plane would be represented by
⮚ D = |ax1 + by1 + cz1 + d|/sqrt(a^2 + b^2 + c^2)
Ex. Problem: Find an equation of the plane that passes through the points P(1,3,2), Q(3,-1,6) and R(5,2,0) (pg. 570)
❖ Be able to visualize graphs of cylinders and quadric surfaces from their equations ⮚ Know that z=x^2 doesn’t involve y so the curve would only intersect at y=0. ❖ Most general equation of a quadric surface is:
⮚ Ax^2 + By^2 + Cz^2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0 ☺
❖ Memorize equations for the 6 surfaces in Table 1 on pg. 578
Ex. Problem: Classify the quadric surface x^2 + 2z^2 – 6x – y + 10 = 0 hint(complete the square) (Pg. 578)
❖ To take the limit of r(t) = <f(t),g(t),h(t)>, take the limit of each separate function: ⮚ Lim r(t) = <lim f(t), lim g(t), lim h(t)
❖ Be able to visualize parametric curves (or space curves)
❖ Be able to find equations representing the intersection points of 2 curves/planes ❖ Know your derivatives and the rules with them (pg. 587)
❖ Know how to integrate (pg. 588)
⮚ Can integrate each component of a vector separately to get the integral of that vector
Ex. Problem: Find a vector function that represents the curve of intersection of the cylinder x^2 + y^2 = 1 and the plane y + z = 2.
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❖ Arc length formula:
⮚ L = integral from a to b of sqrt([f’(t)]^2 + [g’(t)]^2 + [h’(t)]^2) dt
⮚ Or : L = integral from a to b of |r’(t)| dt
❖ Curvature: A parametrization r(t) is called smooth on an interval ‘I’ if r’ is continuous and r’(t) does not equal 0 on I.
⮚ T(t) = r’(t)/|r’(t)|
❖ The curvature of a curve is k= |dT/ds| where T is the unit tangent vector.
⮚ So k(t) = |T’(t)|/|r’(t)|
❖ The curvature of the curve given by the vector function r is k(t) = |r’(t) X r”(t)|/|r’(t)|^3 ⮚ This is probably the most important equation for curvature
❖ Idk if this is necessary but when y = f(x) then:
⮚ K(x) = |f”(x)|/[1 + (f’(x))^2]^(3/2) basically just an easier equation to use when you are finding curvature and y is a function
❖ Normal and Binormal vectors:
⮚ N(t) = T’(t)/|T’(t)|where T(t) is taken from the equation above
⮚ B(t) = T(t) X N(t)
Ex. Problem: Find the curvature of the twisted cubic r(t) = <t,t^2,t^3> at a general point and at (0,0,0). (pg. 595)
Problem 39 on page 599.
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❖ Know how to integrate from a(t) to r(t)
⮚ Use initial conditions and make sure to plug in 0 to see what you get for each component in order to find C
❖ Circular motion:
⮚ General formula: r(t) = acos(wt) i + asin(wt) j
⮚ F(t) = ma(t) = -mw^2(a cos(wt) i + a sin(wt) j) where w = 2pi/T where T = period ❖ Trajectory:
⮚ X = (v0cos (theta)t) and y = (v0 sin (theta)t – ½gt^2)
❖ Planetary Motion:
⮚ F = ma
⮚ F = -((GMm)/r^3)*r
⮚ A = -(GM/r^3)*r
Ex. Problem: A projectile is fired with muzzle speed 150 m/s and angle of elevation 45 degrees from a position 10 m above ground level. Where does the projectile hit the ground, and with what speed? (pg. 603)