••••••MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 50yvLP(x 0 ,y 0 )(a,b)xzP 0 (x 0 ,y 0 ,z 0 )vL(a,b,c)yxFor example, consider a l<strong>in</strong>e L <strong>in</strong> 3-space that passes through the po<strong>in</strong>t P 0 (x 0 ,y 0 ,z 0 ) andis parallel to the nonzero vector v = 〈a,b,c〉. Then L consists precisely of those po<strong>in</strong>tP(x,y,z) for which the vector −−→ P 0 P is parallel to v.zP 0 (x 0 ,y 0 ,z 0 )vP•L(a,b,c)yxIn other words, the po<strong>in</strong>t P(x,y,z) is on L if and only if −−→ P 0 P is a scalar multiple of v, sayThis equation can be written aswhich implies that−−→P 0 P = tv〈x−x 0 ,y −y 0 ,z −z 0 〉 = 〈ta,tb,tc〉x−x 0 = ta, y −y 0 = tb, z −z 0 = tcThus, L can be described by the parametric equationsx = x 0 +at, y = y 0 +bt, z = z 0 +ctA similar description applies to l<strong>in</strong>es <strong>in</strong> 2-space.Theorem 3.14(a) The l<strong>in</strong>e <strong>in</strong> 2-space that passes through the po<strong>in</strong>t P 0 (x 0 ,y 0 ) and is parallel to thenonzero vector v = 〈a,b〉 = ai+bj has parametric equationsx = x 0 +at, y = y 0 +bt (3.22)
MA112 Section 750001: Prepared by Dr.Archara Pacheenburawana 51(b) The l<strong>in</strong>e <strong>in</strong> 3-space that passes through the po<strong>in</strong>t P 0 (x 0 ,y 0 ,z 0 ) and is parallel to thenonzero vector v = 〈a,b,c〉 = ai+bj+ck has parametric equationsx = x 0 +at, y = y 0 +bt, z = z 0 +ct (3.23)Example 3.27 F<strong>in</strong>d parametric equations of the l<strong>in</strong>e pass<strong>in</strong>g through the po<strong>in</strong>t (1,5,2) andparallel to the vector v = 〈4,3,7〉. Also, determ<strong>in</strong>e where the l<strong>in</strong>e <strong>in</strong>tersects the yz-plane.Solution .........Example 3.28 F<strong>in</strong>dparametricequationsof the l<strong>in</strong>eLpass<strong>in</strong>gthrough the po<strong>in</strong>tP(1,2,−1)and Q(5,−3,4).Solution .........Example 3.29 Let L 1 and L 2 be the l<strong>in</strong>es(a) Are the l<strong>in</strong>es parallel?(b) Do the l<strong>in</strong>es <strong>in</strong>tersect?Solution .........L 1 : x = 1+4t, y = 5−4t, z = −1+5tL 2 : x = 2+8t, y = 4−3t, z = 5+tTwo l<strong>in</strong>es <strong>in</strong> 3-space that are not parallel and do not <strong>in</strong>tersect are called skew l<strong>in</strong>es.L<strong>in</strong>e SegmentsSometimes one is not <strong>in</strong>terested <strong>in</strong> an entire l<strong>in</strong>e, but rather some segment of a l<strong>in</strong>e. Parametricequations of a l<strong>in</strong>e segment can be obta<strong>in</strong>ed by f<strong>in</strong>d<strong>in</strong>g parametric equation for theentire l<strong>in</strong>e, then restrict<strong>in</strong>g the parameter appropriately so that only the desired segment isgenerated.Example 3.30 F<strong>in</strong>d parametric equations describ<strong>in</strong>g the l<strong>in</strong>e segment jo<strong>in</strong><strong>in</strong>g the po<strong>in</strong>tsP(1,2,−1) and Q(5,−3,4).Solution .........Vector Equations of L<strong>in</strong>esWe will now show how vector notation can be used to express the parametric equations ofa l<strong>in</strong>e. Because two vectors are equal if and only if their components are equal, (3.22) and(3.23) can be written <strong>in</strong> vector form as〈x,y〉 = 〈x 0 +at,y 0 +bt〉〈x,y,z〉 = 〈x 0 +at,y 0 +bt,z 0 +ct〉