A line L through the origin in R3 can be represented by parametric equations of the form x = at, y = bt, and z = ct. Use these equations to show that L is a subspase of RR3 by showing that if v1=(x1,y1,z1) and v2=(x2,y2,z2) are points on L and k is any real number, then kv1 and v1+v2 are also points on L.

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A line L through the origin in R3 can be represented by parametric equations of the form x = at, y = bt, and z = ct. Use these equations to show that L is a subspase of RR3  by showing that if v1=(x1,y1,z1) and v2=(x2,y2,z2)  are points on L and k is any real number, then kv1 and v1+v2  are also points on L.

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Let v1=(x1,y1,z1)andv2=(x2,y2,z2) be elements of L.Then  {x1=at1y1=bt1z1=ct1and{x2=at2y2=bt2z2=ct2  for some real numbers t1andt2.  let w=v1+v2=(x1+x2,y1+y2,z1+z2).Then  {x1+x2=at1+at2=a(t1+t2)y1+y2=bt1+bt2=b(t1+t2)z1+z2=ct1+ct2=c(t1+t2)  Hence the components of w are of the form x=at,y=bt,z=ctfort=t1+t2.  Now let k be a real number. And consider w=kv1=(kx1,ky1,kz1).Then  {kx1=kat1=a(kt1)ky1=kbt1=b(kt1)kz1=kct1=c(kt1)  Hence the components of w are of the form x=at,y=bt,z=ctfort=kt1.  Then, by the subspace theorem, L is a subspace pf R3.

A line L through the origin in RR^3 can be represented by parametric equations of the form x = at, y = bt, and z = ct.

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