Info Catalog calc: Vector/Matrix Tutorial calc: Vector/Matrix Tutorial calc: Matrix Tutorial

calc: Vector Analysis Tutorial

 
 3.3.1 Vector Analysis
 ---------------------
 
 If you add two vectors, the result is a vector of the sums of the
 elements, taken pairwise.
 
      1:  [1, 2, 3]     2:  [1, 2, 3]     1:  [8, 8, 3]
          .             1:  [7, 6, 0]         .
                            .
 
          [1,2,3]  s 1      [7 6 0]  s 2      +
 
 Note that we can separate the vector elements with either commas or
 spaces.  This is true whether we are using incomplete vectors or
 algebraic entry.  The ‘s 1’ and ‘s 2’ commands save these vectors so we
 can easily reuse them later.
 
    If you multiply two vectors, the result is the sum of the products of
 the elements taken pairwise.  This is called the “dot product” of the
 vectors.
 
      2:  [1, 2, 3]     1:  19
      1:  [7, 6, 0]         .
          .
 
          r 1 r 2           *
 
    The dot product of two vectors is equal to the product of their
 lengths times the cosine of the angle between them.  (Here the vector is
 interpreted as a line from the origin ‘(0,0,0)’ to the specified point
 in three-dimensional space.)  The ‘A’ (absolute value) command can be
 used to compute the length of a vector.
 
      3:  19            3:  19          1:  0.550782    1:  56.579
      2:  [1, 2, 3]     2:  3.741657        .               .
      1:  [7, 6, 0]     1:  9.219544
          .                 .
 
          M-<RET>             M-2 A          * /             I C
 
 First we recall the arguments to the dot product command, then we
 compute the absolute values of the top two stack entries to obtain the
 lengths of the vectors, then we divide the dot product by the product of
 the lengths to get the cosine of the angle.  The inverse cosine finds
 that the angle between the vectors is about 56 degrees.
 
    The “cross product” of two vectors is a vector whose length is the
 product of the lengths of the inputs times the sine of the angle between
 them, and whose direction is perpendicular to both input vectors.
 Unlike the dot product, the cross product is defined only for
 three-dimensional vectors.  Let’s double-check our computation of the
 angle using the cross product.
 
      2:  [1, 2, 3]  3:  [-18, 21, -8]  1:  [-0.52, 0.61, -0.23]  1:  56.579
      1:  [7, 6, 0]  2:  [1, 2, 3]          .                         .
          .          1:  [7, 6, 0]
                         .
 
          r 1 r 2        V C  s 3  M-<RET>    M-2 A * /                 A I S
 
 First we recall the original vectors and compute their cross product,
 which we also store for later reference.  Now we divide the vector by
 the product of the lengths of the original vectors.  The length of this
 vector should be the sine of the angle; sure enough, it is!
 
    Vector-related commands generally begin with the ‘v’ prefix key.
 Some are uppercase letters and some are lowercase.  To make it easier to
 type these commands, the shift-‘V’ prefix key acts the same as the ‘v’
 key.  (General Mode Commands, for a way to make all prefix keys
 have this property.)
 
    If we take the dot product of two perpendicular vectors we expect to
 get zero, since the cosine of 90 degrees is zero.  Let’s check that the
 cross product is indeed perpendicular to both inputs:
 
      2:  [1, 2, 3]      1:  0          2:  [7, 6, 0]      1:  0
      1:  [-18, 21, -8]      .          1:  [-18, 21, -8]      .
          .                                 .
 
          r 1 r 3            *          <DEL> r 2 r 3            *
 
    (•) *Exercise 1.*  Given a vector on the top of the stack, what
 keystrokes would you use to “normalize” the vector, i.e., to reduce its
 length to one without changing its direction?  1 Vector Answer 1.
 (•)
 
    (•) *Exercise 2.*  Suppose a certain particle can be at any of
 several positions along a ruler.  You have a list of those positions in
 the form of a vector, and another list of the probabilities for the
 particle to be at the corresponding positions.  Find the average
 position of the particle.  2 Vector Answer 2.  (•)
 

Info Catalog calc: Vector/Matrix Tutorial calc: Vector/Matrix Tutorial calc: Matrix Tutorial