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Concept 01 / 06Three Dimensional Geometry

Coordinates in Three Dimensions: Distance, Section Formula and Direction Cosines of a Line

Octants and coordinates of a point in space, the distance and section formulas, centroid of a triangle and tetrahedron, direction cosines and direction ratios of a line through two points, the angle between two lines, and projection of a segment on a line.

Figure 1
In three dimensions a point is located by three coordinates (x,y,z)(x, y, z), its signed distances from the yzyz-, zxzx- and xyxy-planes. The three coordinate planes divide space into eight octants, according to the signs of x,y,zx, y, z. The point (x,y,z)(x, y, z) has position vector xi^+yj^+zk^x\hat{i} + y\hat{j} + z\hat{k}, so every result from vector algebra applies directly. Distance: AB=(x2−x1)2+(y2−y1)2+(z2−z1)2AB = \sqrt{(x_{2} - x_{1})^{2} + (y_{2} - y_{1})^{2} + (z_{2} - z_{1})^{2}} (Pythagoras applied twice). The distance of (x,y,z)(x, y, z) from the origin is x2+y2+z2\sqrt{x^{2} + y^{2} + z^{2}}, from the xx-axis is y2+z2\sqrt{y^{2} + z^{2}}, and from the xyxy-plane is ∣z∣|z|. Section formula: the point dividing ABAB internally in the ratio m:nm : n is (mx2+nx1m+n,my2+ny1m+n,mz2+nz1m+n)\left(\frac{mx_{2} + nx_{1}}{m + n}, \frac{my_{2} + ny_{1}}{m + n}, \frac{mz_{2} + nz_{1}}{m + n}\right); externally, replace nn by −n-n. The ratio in which a coordinate plane divides a segment is found by setting that coordinate of the section point to zero: the xyxy-plane divides ABAB in the ratio −z1:z2-z_{1} : z_{2}. The centroid of a triangle is the average of its vertices; the centroid of a tetrahedron is the average of its four vertices. Direction cosines and direction ratios of a line. A line making angles α,β,γ\alpha, \beta, \gamma with the positive axes has direction cosines l=cos⁡αl = \cos\alpha, m=cos⁡βm = \cos\beta, n=cos⁡γn = \cos\gamma, with l2+m2+n2=1l^{2} + m^{2} + n^{2} = 1. Since a line can be traversed in two directions, (−l,−m,−n)(-l, -m, -n) are also direction cosines of the same line. Direction ratios are any triple (a,b,c)(a, b, c) proportional to (l,m,n)(l, m, n). For the line through AA and BB, the direction ratios are (x2−x1,y2−y1,z2−z1)(x_{2} - x_{1}, y_{2} - y_{1}, z_{2} - z_{1}), and dividing by the length ABAB gives the direction cosines. In vector language, (l,m,n)(l, m, n) are the components of a unit vector along the line. Angle between two lines with direction cosines (l1,m1,n1)(l_{1}, m_{1}, n_{1}) and (l2,m2,n2)(l_{2}, m_{2}, n_{2}): cos⁡θ=∣l1l2+m1m2+n1n2∣\cos\theta = |l_{1}l_{2} + m_{1}m_{2} + n_{1}n_{2}| (the dot product of the unit vectors; the absolute value gives the acute angle). With direction ratios (a1,b1,c1)(a_{1}, b_{1}, c_{1}) and (a2,b2,c2)(a_{2}, b_{2}, c_{2}): cos⁡θ=∣a1a2+b1b2+c1c2∣a12+b12+c12a22+b22+c22\cos\theta = \frac{|a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2}|}{\sqrt{a_{1}^{2} + b_{1}^{2} + c_{1}^{2}}\sqrt{a_{2}^{2} + b_{2}^{2} + c_{2}^{2}}}. Perpendicular lines: a1a2+b1b2+c1c2=0a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2} = 0. Parallel lines: a1a2=b1b2=c1c2\frac{a_{1}}{a_{2}} = \frac{b_{1}}{b_{2}} = \frac{c_{1}}{c_{2}}. Also sin⁡2θ=∑(m1n2−m2n1)2\sin^{2}\theta = \sum(m_{1}n_{2} - m_{2}n_{1})^{2}, which comes from the cross product. Projection. The projection of the segment ABAB on a line with direction cosines (l,m,n)(l, m, n) is ∣l(x2−x1)+m(y2−y1)+n(z2−z1)∣|l(x_{2} - x_{1}) + m(y_{2} - y_{1}) + n(z_{2} - z_{1})|. Three points are collinear iff the direction ratios of ABAB and ACAC are proportional. Key Takeaways: - Distance Δx2+Δy2+Δz2\sqrt{\Delta x^{2} + \Delta y^{2} + \Delta z^{2}}; from the xx-axis y2+z2\sqrt{y^{2} + z^{2}}. - Section formula componentwise; the xyxy-plane divides ABAB in ratio −z1:z2-z_{1} : z_{2}. - l2+m2+n2=1l^{2} + m^{2} + n^{2} = 1; direction ratios of ABAB are coordinate differences. - cos⁡θ=∣a1a2+b1b2+c1c2∣∑a12∑a22\cos\theta = \frac{|a_{1}a_{2} + b_{1}b_{2} + c_{1}c_{2}|}{\sqrt{\sum a_{1}^{2}}\sqrt{\sum a_{2}^{2}}}. - Perpendicular: ∑a1a2=0\sum a_{1}a_{2} = 0; parallel: proportional ratios. - Projection of ABAB on a line: ∣lΔx+mΔy+nΔz∣|l\Delta x + m\Delta y + n\Delta z|. - Pitfall: treating direction ratios as direction cosines without normalising. Brain-Teaser 1: Find the ratio in which the yzyz-plane divides the segment joining (−2,4,7)(-2, 4, 7) and (3,−5,8)(3, -5, 8), and the point of division. Solution: On the yzyz-plane x=0x = 0. With ratio k:1k : 1, the xx-coordinate is 3k−2k+1=0\frac{3k - 2}{k + 1} = 0, so k=23k = \frac{2}{3}: the ratio is 2:32 : 3 internally. The point is (0,2(−5)+3(4)5,2(8)+3(7)5)=(0,25,375)\left(0, \frac{2(-5) + 3(4)}{5}, \frac{2(8) + 3(7)}{5}\right) = \left(0, \frac{2}{5}, \frac{37}{5}\right). A positive kk means internal division, which is consistent with the endpoints having xx-coordinates of opposite signs. Brain-Teaser 2: A line makes angles α,β,γ,δ\alpha, \beta, \gamma, \delta with the four body diagonals of a cube. Show that cos⁡2α+cos⁡2β+cos⁡2γ+cos⁡2δ=43\cos^{2}\alpha + \cos^{2}\beta + \cos^{2}\gamma + \cos^{2}\delta = \frac{4}{3}. Solution: The body diagonals of the unit cube have direction ratios (1,1,1)(1, 1, 1), (−1,1,1)(-1, 1, 1), (1,−1,1)(1, -1, 1) and (1,1,−1)(1, 1, -1), each with length 3\sqrt{3}. For a line with direction cosines (l,m,n)(l, m, n): cos⁡α=l+m+n3\cos\alpha = \frac{l + m + n}{\sqrt{3}}, and similarly with sign changes. Summing the squares: 13[(l+m+n)2+(−l+m+n)2+(l−m+n)2+(l+m−n)2]=13×4(l2+m2+n2)=43\frac{1}{3}[(l + m + n)^{2} + (-l + m + n)^{2} + (l - m + n)^{2} + (l + m - n)^{2}] = \frac{1}{3} \times 4(l^{2} + m^{2} + n^{2}) = \frac{4}{3}, because all the cross terms cancel. The result is independent of the line, a neat use of l2+m2+n2=1l^{2} + m^{2} + n^{2} = 1. Brain-Teaser 3: Show that the points A(2,3,−4)A(2, 3, -4), B(1,−2,3)B(1, -2, 3) and C(3,8,−11)C(3, 8, -11) are collinear. Solution: Direction ratios of ABAB: (−1,−5,7)(-1, -5, 7); of ACAC: (1,5,−7)(1, 5, -7). These are proportional (with factor −1-1), and the lines share the point AA, so the three points are collinear, with AA the midpoint of BCBC (indeed 1+32=2\frac{1 + 3}{2} = 2, −2+82=3\frac{-2 + 8}{2} = 3, 3−112=−4\frac{3 - 11}{2} = -4). Proportional direction ratios alone give parallel lines; the shared point is what makes them the same line.

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Coordinates in Three Dimensions: Distance, Section Formula and Direction Cosines of a Line — DPP

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