if l P v Q w how must P Q v and w be related note that l P

if l = P + [v] =Q +[w], how must P, Q, v and w be related? note that l= P+[v] be a line with unit vector v then, [v] is called the span of v, l is a line, P is a point , Q is a point, and [w] is called the span of w

Solution

Given l=P+[v] is a line where [v] is a span of a unit vector v, we can write [v]=cv, c is a scalar which spans v.

when c=0, we get P+[v] =P+cv=P+0=P. i.e the point P lies on a line l.

Similarly

l=Q+[w] is a line where [w] is a span of a vector w, we can write [w]=kw, k is a scalar which spans w.

when k=0, we get Q+[w] =Q+kw=Q+0=Q. i.e the point Q lies on a line l.

From these two points, we understood that the points P and Q lies on the line l. i.e Q=P+nv; n is a scalar.

since P and Q are on same line l, we can say that [v] and [w] are same by saying that directions of v and w are same but magnitudes may be different.

Since the directions of v and w are same, we can write the vector w is a scalar combination of unit vector v,

i.e w=dv, where d is a scalar such that ||w||=|d|.

Finally we can write Q+[w]=(P+nv)+kw = (P+nv)+kdv = P + (n+kd)v = P + [v], both represents the same line l.