If an area is multiplied by a
length, is the resultant quantity a vector quantity or a scalar quantity?
All right, so we’re talking about
multiplying an area by a length. And we could imagine, say, that
this is our area and that this here is our length. When we multiply this area by this
length, we’re going to get some result, and we want to know whether that’s a vector
quantity or a scalar quantity. Here’s the difference between the
two. A scalar quantity, or scalar for
short, has a magnitude only, while a vector quantity, or a vector, has both
magnitude and direction. So, to figure out whether our area
multiplied by a length is a scalar or a vector, we’ll need to see whether it has a
magnitude only or a magnitude along with a direction.
The product of our area and our
length is this volume 𝑉. And so now we ask ourselves, does
this volume have a magnitude only or does it have a magnitude along with a
direction? Put that way, we can see that this
volume does have a magnitude, that is, a size, that’s equal to whatever 𝐴 times 𝐿
is. But there’s no particular direction
in which this volume or any volume points. So here, we do have a magnitude,
but we don’t have a direction. Based on our definitions of scalar
and vector quantities, that then answers our question. If an area is multiplied by a
length, the resultant quantity is a scalar quantity.