Calculus in 3D. Geometry, vectors, and multivariate calculus by Nitecki Z.

By Nitecki Z.

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Extra resources for Calculus in 3D. Geometry, vectors, and multivariate calculus

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In particular, the locus of a “linear” equation in the three rectangular coordinates x, y and z Ax + By + Cz = D is a plane, not a line. 14). This formula separates two pieces of geometric data which together determine a line: the slope reflects the direction (or tilt) of the line, and then the y-intercept distinguishes between the various parallel lines with a given slope by specifying a point which must lie on the line. 6 Then, to pick out one among all the lines parallel → v , we specify a basepoint P0 (x0 , y0 , z0 ) through which the line is to − required to pass.

It follows that ℓ1 and ℓ2 are distinct, but parallel, lines. Now, consider a third line, ℓ3 , given by x = 1 +3t y = 2 +t z = −3 +t. 36 CHAPTER 1. COORDINATES AND VECTORS We read off its direction vector as → − − → → → v 3 = 3− ı +−  + k which is clearly not a scalar multiple of the other two. This tells us immediately that ℓ3 is different from both ℓ1 and ℓ2 (it has a different direction). Let us ask whether ℓ2 intersects ℓ3 . It might be tempting to try to answer this by looking for a solution of the vector equation − → → p 2 (t) = − p 3 (t) but this would be a mistake.

25: Barycentric Coordinates The numbers λi are called the barycentric coordinates of P with respect to A, B and C. Show that P lies on an edge of the triangle precisely if one of its barycentric coordinates is zero. Barycentric coordinates were introduced (in a slightly different form) by August M¨obius (1790-1860)) in his book Barycentrische Calcul (1827). 8 10. Find a line that lies entirely in the set defined by the equation x2 + y 2 − z 2 = 1. History note: 11. Heath [27, pp. 26). This was proved by vector methods in Exercise 8.

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