What is really meant by the product of two vectors?

Question

You may find it silly but I want to know what the product of two vectors really mean (dot and cross product). For instance the product of two numbers lets say 2 and 3 means that we have two of threes or three of twos which is equal to 6. But what does it mean to multiply two numbers with directions sticking to them?

Answer 1

This question was migrated from Physics SE. This answer was given from a physics perspective.

• You can add a force to a force. Mathematically, you add them element by element. Geometrically, you draw them after one another. It intuitively makes somewhat sense because both effects are taking place simultaneously from a physics perspective.

• But multiplying a force with a force is meaningless. This does not happen in the real world.

For any sense of multiplication to make sense we thus consider what is meaningful in physics.

• Oftentimes we need only the parallel component values of a vector multiplied together. Such as when pushing on a cart on a track , where only the force component that is along with the cart displacement actually helps the cart moving and does work on the cart, $W=\vec F\cdot\vec r$. Any sideways component has no influence. The dot product is a mathematical invention that multiplies the parallel component values of two vectors together: $$\vec a \cdot \vec b=ab_\parallel=a_\parallel b=ab\cos(\theta).$$

• Other times we need not the parallel components but the perpendicular component values multiplied. Such as when using a wrench to tighten a screw. You pull the wrench to induce a torque, $\vec \tau=\vec r\times\vec F$. Any force component along with the wrench arm makes no difference - only force components that are perpendicular to the wrench arm make it turn and induce a torque. The cross product is a mathematical invention that multiplies the perpendicular components of two vectors: $$|\vec a \times \vec b|=ab_\perp=a_\perp b=ab\sin(\theta).$$

On top of this, someone has also considered that maybe it would make sense to give torque a "direction". Meaning, it might make sense to consider it as a vector. Thus, the cross product is actually defined to also have a direction (which in fact isn't very physical - torque is a so-called pseudovector), contrary to the dot product.

As you can see, different mathematical operations can be invented for different physical needs. (Or vice versa, with different physical needs you will pick and choose among the mathematical operations that have been defined/invented.) Which one would make more sense as "vector multiplication"? That is not an unambiguous question - in fact, it is meaningless, but also not important. You might even see yet other definitions of products that you just as well could consider "vector multiplication", such as the scalar tripple product. The essense is that multiplication and addition feel intuitive for scalars whereas they do not for vectors and other mathematical objects. But, again, it doesn't matter. As long as we have the operations we need for the needs we have.

Answer 2

In general, given a set $A$, a product is a function that takes two elements in $A;$ $f(a_1, a_2) = a_1 \cdot a_2.$ and spits out something else. Suppose $A$ is a vector space.

Then if the term product is used, this generally means the function is bilinear. To be bilinear means that the functions has the following properties:

1. $f(a_1, a_2 + a_3) = f(a_1, a_2) + f(a_1, a_3)$ and vice-versa.
2. $f( \lambda a_1, a_2) = \lambda f(a_1, a_2) = f(a_1, \lambda a_2)$

Any function on a vector space satisfying those requirements deserves to be called a product, because algebraically it behaves similarly to multiplication in the real numbers.

A product which has the following additional property is called commutative

• $f(a_1, a_2) = f(a_2, a_1)$

A product which has the following additional property is called anti-commutative

• $f(a_1, a_2) = -f(a_2, a_1)$

Some products are also associative.

The dot product is a commutative product on a vector space with a finite number of dimensions that returns a scalar, and generally is useful because it captures some properties of Euclidian geometry.

The cross product is an anticommutative product on only three dimensions that returns a vector perpendicular to the other two in Euclidian geometry, along with other properties related to Euclidian geometry.

Many of the answers here claim that there is no “one” or “correct” way to multiply vectors. Arguably, in Euclidian geometry (in other words, given a metric) the geometric product is the most general and “right” way to multiply vectors.

Answer 3

Well, briefly speaking, dot product means the projection of one vector to the other one:

$\vec a.\vec b = \left|\vec b\right| (\vec a.\hat b) = b.a cos\theta = b$ times the projection of $\left|\vec a\right|$ on / along $\hat b$.

On the other hand, ceoss product means you are calculating a vector perpendicular to the plane, where both the given vector resides on.

$\vec a×\vec b$ is perpendicular to both of $\vec a$ and $\vec b$.

You may also interoret the cross product as the area (vector) of a parallelogram whose two adjacent sides are represented by the given vectors. You can easily prove that by considering $\vec a×\vec b = ab.sin\theta.\hat n$, where $\hat n$ has been taken following the right thumb rule.

Answer 4

When $\vec c=\vec a \times \vec b$, it means that

1-$\vec c$ is perpendicular to both $\vec a$ and $\vec b$

2-$|\vec c|$ is twice the area of the triangle made by two adjacent vectors $\vec a$ and $\vec b$.

3-$|\vec c|$ is the area of parallelogram made by two adjacent vectors $\vec a$ and $\vec b$.

4-$|\vec c|$ is the area of each quadrilateral made by two diagonal vectors $\vec a$ and $\vec b$.

5-$\vec c/b$ is component of $\vec a$ perpendicular to $\vec b$.