(1) Consider the parametric vector equation→x = ⟨1 + t, 2 + t, 3 − t⟩.What position will →x be at when t = 0? What about t = 1?
(2) Write the parametric vector equation of a line that passes through the point (1, 3, 4) at t = 0, having velocity vector ⟨−1, 2, −1⟩.
(3) To find the distance between a point, Q, and a line, l, we need to construct the vector P→Q where P is any point on line l.
Suppose l is given parametrically by→x = ⟨1 − 2t, 5 + 3t, 4 − t⟩.
(a) What is an obvious choice for point P ?
(b) If Q = (1, 6, 5), determine P→Q and find its cross product with the velocity vector, →v, for the line l.
(c) Since the distance between Q and l is given by |P→Q| • | sin θ| (where θ is the angle between P Q and →v), we can find the distance using the cross product in the previous problem. Do it.
(4) Suppose a line is given parametrically by →x = ⟨2t − 1, 7t, 6t + 5⟩.Find the distance between this line and the point (2, 5, 13).
(5) When we want to find the distance between a point and a plane, we use a dot product with the normal vector for the plane. Consider the plane whose scalar equation is 2x + 3y − z = 10.The ingredients we’ll need for computing the distance from some point Q
to this plane are:
(a) A normal vector for the plane. ()The coordinates of this vector are literally visible in the equation.) What is it?
(b) A second point that lies in the plane. Often this can be determined by setting two of the variables to 0. Since we just need any point in the plane, it’s a good idea to choose which variables to set to 0, in such a way that the calculation is easy. Find a nice point that lies in this plane.
(6) Find the distance from the plane given in the previous problem to the pointQ = (3, 4, 5).
(7) Two planes will generally intersect in a line. If their normal vectors point in the same direction they may be parallel. Parallel planes can even be literally the same, even though their equations appear to be different! But, if the normal vectors are not multiples of one another, the planes will intersect in a line.
Identify the following situations (there is one of each) Parallel planes, Iden- tical planes, Planes that intersect in a line.
(a) 3x + 4y − 5z = 9 and −6x − 8y + 10z = −18
(b) 3x − 2y + z = 2 and 4x + y − z = 4
(c) 2x + 4y + 6z = 9 and 2x + 4y + 6z = 12
(8) When two planes are not parallel, a vector that lies in their intersection will be perpendicular to both of their normal vectors. This allows us to use a cross product to find a vector that can be used as the velocity vector in finding a parametric description for the line of intersection.
Find a parametric equation for the line of intersection of x − y + 2z = 5 and 2x + 3y − z = 5.Hint: (1, 2, 3) is a point that lies in both planes.
(1) Generalized cylinders have equations that don’t involve one of the variables. They will run parallel to the missing variable’s axis and have a cross-section determined by the equation.
For example y x2 = 0 is a generalized cylinder with a parabolic cross- section that runs parallel to the z-axis.
Identify each of the following cylinders.
(a) x2 + z2 = 4
(b) y2 − z2 = 1
(c) x − z2 = 0
(d) y2 + x2 = 9
(e) y3 − x = 0
(2) A way to get started on figuring out the graph of an arbitrary quadratic polynomial in 3 variables is to determine where (or if) it intersects the axes. To do this you set the other two variables to 0 and solve. These points are the intercepts – there may be 2, 1, or 0 intercepts on each axis.What are the intercepts of the following?
(a) x2 + y2 − z = 4
(b) x2 − y2 − z = 9
(c) x2 − y2 + z2 = 16
(d) y2 + x2 = 9
(e) (x/3)2 + (y/4)2 + (z/2)2 = 1
(3) Occasionally, the process of finding the intercepts on an axis fails spectacularly when the graph contains the entire axis. Which axis is wholly contained in the graph of y − z2 = 0?
(4) The next step in identifying a quadric surface is to determine the traces – the plane curves where the surface intersects the coordinate planes. To do this you need to be able to recognize the forms of plane curves in two variables. What are the following? (Draw sketches.)
(a) x2 + y2 = 4
(b) x2 + y2 = −4
(c) x2 − y = 0
(d) y2 − x2 = 1
(e) x2 − y2 = 1
(f) x2 − y2 = 0
(5) What are the traces of the following? Specify a curve (which may be empty or non-existent!) for each of the xy, xz and yz planes. (You’ll need to set one variable to zero – for instance to find the trace in the xz plane, set y = 0.)
(a) x2 + y2 + z2 = 4
(b) x2 − y2 − z = 9
(c) x2 + y2 − z2 = 1
(d) y2 + x2 = 9
(e) x2 − y2 − z2 = 1
(6) Suppose you determine that a graph has traces in the xz and yz planes that are parabolas and that it’s trace in the xy plane is a circle. Which kind of quadric surface is it?
(7) A quick way to distinguish the two kinds of hyperboloids (1-sheet and 2-sheet) is that the hyperboloid of two sheets has one trace that is non-existent. For instance x2 + y2 z2 = 1 misses the xy plane entirely. In looking at the equation (with z 0 we see a sum of squares that needs to be negative which is impossible.
Each of the following is a hyperboloid of two sheets. Which coordinate planes have these “empty” traces?
(a) x2 + y2 − z2 = −1
(b) x2 − y2 + z2 = −1
(c) x2 − y2 − z2 = 1
(8) Describe the object in three-dimensional space defined by the equation 3x + 2y − z = 11.
(9) Describe the object in three-dimensional space defined by the equation x2 +y2 + z2 = 25.
(10) Describe the object in three-dimensional space defined by the equation x2 y = 0.
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