Figure 10.50: Sketching the lines from Ex-ample 346.Chapter 10VectorsThe next two examples invesƟgate these possibiliƟes.Example 346Comparing linesConsider linesℓ1andℓ2, given in parametric equaƟon form:ℓ1:x=1+3ty=2-tz=tℓ2:x=-2+4sy=3+sz=5+2s.Determine whetherℓ1andℓ2are the same line, intersect, are parallel, or skew.SÊ½çã®ÊÄWe start by looking at the direcƟons of each line. Lineℓ1has the direcƟon given by⃗d1=⟨3,-1,1⟩and lineℓ2has the direcƟon given by⃗d2=⟨4,1,2⟩. It should be clear that⃗d1and⃗d2are not parallel, henceℓ1andℓ2are not the same line, nor are they parallel. Figure 10.50 verifies this fact (wherethe points and direcƟons indicated by the equaƟons of each line are idenƟfied).We next check to see if they intersect (if they do not, they are skew lines).To find if they intersect, we look fortandsvalues such that the respecƟvex,yandzvalues are the same. That is, we wantsandtsuch that:1+3t=-2+4s2-t=3+st=5+2s.This is a relaƟvely simple system of linear equaƟons. Since the last equaƟon isalready solved fort, subsƟtute that value oftinto the equaƟon above it:2-(5+2s) =3+s⇒s=-2,t=1.A key to remember is that we havethreeequaƟons; we need to check ifs=-2,t=1 saƟsfies the first equaƟon as well:1+3(1)̸=-2+4(-2).It does not. Therefore, we conclude that the linesℓ1andℓ2are skew.Example 347Comparing linesConsider linesℓ1andℓ2, given in parametric equaƟon form:ℓ1:x=-0.7+1.6ty=4.2+2.72tz=2.3-3.36tℓ2:x=2.8-2.9sy=10.15-4.93sz=-5.05+6.09s.Determine whetherℓ1andℓ2are the same line, intersect, are parallel, or skew.Notes:608