Letr = (x, y, z) denote a position vector with length r = px2+ y2 + z2
and c is a constant (vector) Determine atvira (c x r))
Answers
In the previous chapters we have considered real functions of several (usually two) variables f : D → R,
where D is a subset of R
n, where n is the number of variables. These are scalar-valued functions in the sense
that the result of applying such a function is a real number, which is a scalar quantity. We now wish to
consider vector-valued functions f : D → R
m. In principal, m can be any positive integer, but we will only
consider the cases where m = 2 or 3, and the results of applying the function is either a 2D or 3D vector.
4.2 Parametric equations of curves
The simplest type of vector-valued function has the form f : I → R
2
, where I ⊂ R. Such a function returns
a 2D vector f(t) for each t ∈ I, which may be regarded as the position vector of some point on the plane.
For example, recall the Section Formula from Level 1. This states that the position vector of any point
P on the line through points A and B is
p =
αa + βb
α + β
,
for any scalars α, β. If we define t = β/(α + β), then this may be rewritten as
p(t) = (1 − t)a + tb.
As t changes, we get different points on the line through A and B and in particular, p(0) = a and p(1) = b.
We may think of p as a vector-valued function
p: R → R
3
,
the image of which is the whole of the line, or
p: [0, 1] → R
3