0
$\begingroup$

I'm beginning to learn some vector calculus, and I am slightly confused about the textbook's explanation of planes spanned by two vectors.

They said for example that the xy plane is an example of the plane spanned by i and j. This concept makes sense to me, but I'm just wondering if a plane needs to necessarily be defined by two vectors perpendicular to each other like the xy , yz , and xz planes.

$\endgroup$
3
$\begingroup$

No. For example, the vectors $(1,0)$ and $(1,1)$ also span the $xy$ plane. You only need two independent vectors to span the plane.

$\endgroup$
1
$\begingroup$

The two vectors need not be perpendicular, but they are not allowed to be parallel to (i.e., scalar multiples of) each other in order to span a plane. For example, the vectors $(1, 0)$ and $(2, 1)$ span the $xy$ plane, but the vectors $(1,0)$ and $(2,0)$ do not.

The key idea is whether you can reach every point in the plane by forming a linear combination, $a_{1}v_{1} + a_{2}v_{2}$, of your two vectors, where $a_{1}$ and $a_{2}$ are scalars (i.e., real numbers, in the context of this question). Note that the vectors $(1,0)$ and $(2,0)$ do span a horizontal line through the origin, but you could achieve the same result with just one of these vectors: the other is redundant for this purpose.

(See also the concept of linear independence. You could well meet it very soon in your multivariable/vector calculus course or in a linear algebra course.)

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.