The dot or scalar product of two vectors, a and b,
is the product of their lengths times the cosine of the angle between them.
This is usually written as either ab or (a, b).
Thus if we take aa we get the square of the length of a.
This product (and the next as well) is linear in either argument (a or b), by which we mean that for any number c we have
c(a, b) = (ca, b) = (a, cb)
and for any vectors a, d and b we have
(a + d, b) = (a, b) + (d, b)
(a, b + d) = (a, b) + (a, d)
This innocent looking fact is very important; it means that if we know all the dot products among some set of vectors, say i, j and k, and can express a and b each as sums of these, then we can read off the dot product of a and b.In particular, if we choose i, j and k to be mutually perpendicular and each of unit length, then
ii = jj = kk = 1
ij = jk = ki = 0
For real vectors, ab
is always the same as ba.
(when complex vectors are defined this is not usually so; instead these two
products are complex conjugates of one another.)
a = (a, i)i + (a, j)j + (a, k)k
b = (b, i)i + (b, j)j + (b, k)k
(a, b) = (a, i)(b, i) + (a, j)(b, j) + (a, k)(b, k)
The square of the magnitude of a is aa, and the direction between two segments can be read off from the fact that the cosine of the angle between them is their dot product divided by the product of their magnitudes, if you know their dot product and their magnitudes.
With dot products you can easily compute the (minimum) distance between a point and a plane , and the distance between a point and a line.
You can prove the law of cosines.
Computation of dot products
1. Its magnitude is area of parollelogram determined by vectors =
2. Its sign changes if the order is reversed: thus
ab = - ba and so aa = 0
3.Conventionally defined so that
ij = k
jk = i,
ki = j (and so ji = - k, etc...)
4. The vector product is orthogonal to either factor:
aba = abb = 05. It can be computed in rectangular coordinates by multiplying out all 9 terms in the product using the distributive law and using the properties of products of i, j and k above to simplify.
6. The same result comes from computing the determinant of the matrix for which the components of a are the first row, those of b the second, and i, j, k the third.
note: good students make careless errors in evaluating cross products 1/3 of the time they try to do so on exams!
Computation of cross products
properties of the double cross a(bc):
1. It is a vector.2. It is orthogonal to a and to the normal to b and c, and hence it is in the plane of b and c, hence it is a linear combination of b and c that is orthogonal to a.
3. It changes sign if b and c are interchanged.
4. If we write a(bc)
as a linear combination of b and c, the coefficient of b
must be linear in a and c;
It turns out to be (a, c); by symmetry the coefficient of c is -(a, b).
a(bc) = (a, c)b - (a, b)c.(This is easy to remember except for the sign; its best to check the sign when you use this formula by trying it on an example you know: a = b = i, c = j).
Computation of double cross products
properties of the mixed double product a(bc)
1. It is a number, not a vector.
2. Its magnitude is the volume of the parallopiped defined by the three vectors.3. Its sign depends on the cyclic order of the vectors: check it by substituting a = i, b = i and c = j, and using what you know about the dot and cross products of these.
4. It can be computed by evaluating the determinant of the matrix whose rows are, in order, the components of a, b, and c.
do not confuse these two products!
Geometric applications of double products
Computation of mixed double products