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1.in scalar multiplication we know only one product like, 2*3 or anything like that but why two different product style available in vector algebra?is this no weird?

$A=2i+3j$

$B=5i+3j$

$A*B=?$

why cant i write this as $A*B=10i+9j$ ?

when we are adding two vectors just simple adding magnitudes but multiplication is so bizzar, isn't it?

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  • $\begingroup$ What's the geometry here? How does the vector $A*B$ (in your notation) relate to the vectors $A$ and $B$? $\endgroup$
    – Angina Seng
    Sep 7, 2020 at 13:21
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    $\begingroup$ @gray Your definition of a product is also one that is occasionally used. It is called the Hadamard product or the Schur product, and is usually written as $A \circ B$. $\endgroup$
    – Ben Grossmann
    Sep 7, 2020 at 13:25
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    $\begingroup$ @gray The problem is that all of these products are sometimes useful, and all of them generalize the usual idea of multiplication in some way. Because there isn't one product that is useful in all circumstances, it does not makes sense to say that any one of these is "the" product $AB$. $\endgroup$
    – Ben Grossmann
    Sep 7, 2020 at 13:28
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    $\begingroup$ The point-wise product IS what you do in, say, the direct sum of rings -- but linear algebra is closely tied to the geometry of vectors, in which dot and cross products are much more useful (and therefore more common). In a sense, the problem is that one encounters linear algebra before abstract ring theory, so the point wise ("common sense") product doesn't 't come up until later. $\endgroup$
    – Ned
    Sep 7, 2020 at 13:46
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    $\begingroup$ One other thought -- the point-wise product has some nasty properties, like lots of $0$ divisors -- which might be worth pointing out in an intro linear algebra course as a reason why it is not the product we want to work with, despite the "obvious" definition. $\endgroup$
    – Ned
    Sep 7, 2020 at 13:53

2 Answers 2

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In 2D, they are not so weird as you think. Write the vectors as complex numbers, $2+3i$ and $5+3i$ (where $i$ is the usual imaginary number, not your $\vec i$). The "ordinary" product is

$$2\cdot5-3\cdot3+(2\cdot3+3\cdot5)i,$$

very close to the product of the first number by the conjugate of the second,

$$2\cdot5+3\cdot3+(2\cdot3-3\cdot5)i,$$

where you recognize both the dot- and the cross-products !

By the way, you say that for addition we are just "adding magnitudes", but this is not true, the magnitude of the sum can be smaller than the sum of the magnitudes.

This whole "complication" (which is in fact a wonderful source of richness) comes form the fact that vectors need not have the same direction and do not add/multiply like scalars.

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  • $\begingroup$ ok i get your point .but what about the direction in this products?why cross product directed perpendicular to the plane? $\endgroup$
    – gray
    Sep 8, 2020 at 15:29
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Those two products represent two different things, both of them useful in the right contexts, that's why the two definitions exist (among others)

The cross product $A \times B$ is a vector whose length is the product of the length of vectors $A$ and $B$ and whose direction is perpendicular to both of them. In other words, it's the normal vector to a plane formed by straight lines in the directions of $A$ and $B$ respectively.

The dot product can be undersood as a product that takes somehow takes into account the angle between $A$ and $B$ (for example, work is the dot product of force and displacement). $A \cdot B$ will be at its biggest if $A$ and $B$ are parallel but will turn to $0$ if they are perpendicular

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