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What are the biaxial planes in Cartesian xyz space
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The biaxial planes are the xy plane, the xz plane, and the yz plane Each plane is perpendicular to the other two, and all three intersect at the origin The biaxial planes are defined by pairs of axes as follows: The xy plane contains the axes for variables x and y The xz plane contains the axes for variables x and z The yz plane contains the axes for variables y and z
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In Cartesian xyz space, a point can always be denoted as an ordered triple in the form (x,y,z) What do the x, y, and z coordinates represent geometrically
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We can think of this situation in two different ways First, we can use the notion of a point s projection We get the projection of a point onto an axis by drawing a line from the point to the axis, and making sure that the line intersects that axis at a right angle That way, the coordinates and projection points are related as follows: The x coordinate represents the point s projection onto the x axis The y coordinate represents the point s projection onto the y axis The z coordinate represents the point s projection onto the z axis We can also think of the x, y, and z values for a particular point in terms of perpendicular displacements from the biaxial planes as follows: The x coordinate is the point s perpendicular displacement (positive, negative, or zero) from the yz plane The y coordinate is the point s perpendicular displacement (positive, negative, or zero) from the xz plane The z coordinate is the point s perpendicular displacement (positive, negative, or zero) from the xy plane
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What semantical distinction should we keep in mind when we talk about points in terms of ordered triples
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An ordered triple represents the coordinates of a point in three-space, not the geometric point itself Informally, the ordered triple is the name of the point We can talk about the ordered triple as if it were the actual point, as long as we re aware of the technical difference between the object and its name
Question 7-7
How can we find the distance of a point from the origin in Cartesian xyz space
Answer 7-7
Suppose we name the point Q, and assign it the coordinates Q = (xq,yq,zq) If we call the distance between Q and the origin by the name dq, then dq = (xq2 + yq2 + zq2)1/2 This distance is always defined, it s always unique (unambiguous), it s never negative, and it doesn t depend on whether we go from the origin to the point or from the point to the origin
Question 7-8
How can we find the distance between two points in Cartesian xyz space
Part One Answer 7-8
Let s call the points and their coordinates S = (xs,ys,zs) and T = (xt,yt,zt) where each coordinate can range over the entire set of real numbers If we go from S to T, the distance between the points is dst = [(xt xs)2 + ( yt ys)2 + (zt zs)2]1/2 If we go from T to S, the distance is dts = [(xs xt)2 + ( ys yt)2 + (zs zt)2]1/2 This distance is always defined and unique It s never negative, and it doesn t depend on which direction we go Therefore dst = dts
Question 7-9
How can we find the midpoint of a line segment connecting two points in Cartesian xyz space
Answer 7-9
Let s call the points and their coordinates P = (xp,yp,zp) and Q = (xq,yq,zq) We can call the midpoint M, and say that its coordinates are M = (xm,ym,zm) Given this information, the coordinates of M in terms of the coordinates of P and Q are (xm,ym,zm) = [(xp + xq)/2,( yp + yq)/2,(zp + zq)/2] This midpoint is always defined, it s always unique, and it doesn t depend on which direction we go
Review Questions and Answers Question 7-10
Suppose that we have two points in Cartesian xyz space where all three pairs of corresponding coordinates are negatives of each other Where is the midpoint of a line segment connecting these two points
Answer 7-10
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