Part 4: The Span of Two Nonparallel Vectors
A linear combination of nonparallel vectors u
and v is a vector of the form
where a and b are numbers. The set of all linear combinations of 2
nonparallel vectors u and v is called the span
of u and v.
Moreover, if u and v are parallel to given plane P,
then the plane P is said to be spanned by u and v.
EXAMPLE 7 Find the equation of the plane through the point P_{1}( 0,0,0) spanned by the vectors u =
á1,2,1
ñ and v =
á 3,1,2
ñ
Solution: The normal to the plane is n = u×v, which is
n = u×v =
á 1,2,1
ñ ×
á 3,1,2
ñ =
á 5,5,5
ñ 

Definition 4.1 then implies that the equation of the plane is
5( x0) +5( y0) 5( z0) = 0 

Solving for z then produces the functional form z = yx.
Finally, since r×( u×v)
is orthogonal to u×v, the vector r×( u×v) must be in the span of u and v. That is, there must be scalars a and b such that r×(u×v) = au+bv.
Also, since r×( u×v) is orthogonal to r, we have
r·( au+bv) = 0 or ar·u = br·v 

Indeed, it is straightforward to show that a = r·v and b = r·u , thus leading to the triple vector product identity:
r×( u×v) = ( r·v) u( r·u) v 
 (4) 
Exercises 4144 will provide additional insights into the interpretation and
application of (4).
EXAMPLE 8 Verify (4) for the vectors
u =
á 1,2,0
ñ , v =
á1,1,0
ñ , and r =
á 3,2,1
ñ 

Solution: Since u×v =
á0,0,1
ñ , the left side of (4) is
r×( u×v) =
á3,2,1
ñ ×
á 0,0,1
ñ =
á2,3,0
ñ 

Notice that u, v, and r×( u×v) are parallel to the xyplane. The right side of (4) is


(
á 3,2,1
ñ ·
á1,1,0
ñ ) u(
á3,2,1
ñ ·
á 1,2,0
ñ ) v 
 

 

 

 


