13.7 The dot product by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.
vCalc Companion Formulas | |
vCalc Formulary | 13.7 The dot product |
`W=Fd` | Work |
`W=F.cos(?).dx` | Work |
Up until now, we have not found any physically useful way to define the multiplication of two vectors. It would be possible, for instance, to multiply two vectors component by component to form a third vector, but there are no physical situations where such a multiplication would be useful.
The equation `W=|F||d|cos?` is an example of a sort of multiplication of vectors that is useful. The result is a scalar, not a vector, and this is therefore often referred to as the scalar product of the vectors `F` and `d`. There is a standard shorthand notation for this operation,
`A*B = |A| |B| costheta `, [definition of the notation `A.B`; `theta` is the angle between vectors `A` and `B`]
and because of this notation, a more common term for this operation is the dot product. In dot product notation, the equation for work is simply
The dot product has the following geometric interpretation:
`A?B=|A|("component of " B " parallel to " A)`
`=|B|("component of " A " parallel to " B)`
The dot product has some of the properties possessed by ordinary multiplication of numbers,
`A*B =B*A`
`A*(B+C)=A*B+A*C`
`(cA)*B=c(A?B)`,
but it lacks one other: the ability to undo multiplication by dividing.
If you know the components of two vectors, you can easily calculate their dot product as follows:
`A?B=A_xB_x+A_yB_y+A_zB_z.`
(This can be proved by first analyzing the special case where each vector has only an `x` component, and the similar cases for `y` and `z`. We can then use the rule `A?(B+C)=A?B+A?C` to make a generalization by writing each vector as the sum of its `x`, `y`, and `z` components. See homework problem 17.)
If we take the dot product of any vector b with itself, we find
`b?b=(b_x x? +b_y? +b_z? )?(b_x x? +b_y? +b_z? )`
`=b_x^2+b_y^2+b_z^2`
so its magnitude can be expressed as `|b|=sqrt(b.b)`,
We will often write `b^2` to mean, `b*b` when the context makes it clear what is intended. For example, we could express kinetic energy as `(1"/"2)m|v|^2`, `(1"/"2)mv?v`, or `(1"/"2)mv^2`. In the third version, nothing but context tells us that `v` really stands for the magnitude of some vector `v`.
? A mule pulls a barge with a force `F=(1100 N)x? +(400 N)? `, and the total distance it travels is `(1000 m)x?`. How much work does it do?
? The dot product is `1.1×10^6 N?m=1.1×10^6 J`.
13.7 The dot product by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.