Answer
yxz(0,0,0) (0,1,0)(0,0,1)(2,−1,1)(−1,0,0)(1,0,0)(1,0,−1)11.5 Distance in Cartesian coordinates
Look again at Figure 11.6. The distance,r, from the originOto the pointP(x 1 ,y 1 ,z 1 )
is given by
r=OP=√
x^21 +y 12 +z 12This can be seen by applying Pythagoras’ theorem (154
➤
)twice:OP^2 =ON^2 +PN^2 =OQ^2 +QN^2 +PN^2 =x 12 +y^21 +z^21In general the distance between two pointsP(x,y,z),P′(x′,y′,z′)is given by:
PP′=√
(x′−x)^2 +(y′−y)^2 +(z′−z)^2which again follows from Pythagoras’ theorem.
Problem 11.1
Calculate the distance between the two pointsP.−1, 0, 2/andP′.1, 2, 3/.The distance is given by substituting the coordinates into the above expression forPP′:
PP′=√
(− 1 − 1 )^2 +( 0 − 2 )^2 +( 2 − 3 )^2=√
9 = 3