ben green
(Ben Green)
#1
2. The magnitude of a vector is a nonnegative scalar quantity. The magnitude of a
vector B is denoted by the symbols B or |B|.
3. A vector B is equal to zero only if its magnitude is zero. A vector whose mag-
nitude is zero is called the zero or null vector and denoted by the symbol 0.
4. Multiplication of a nonzero vector Bby a positive scalar mis denoted by mBand
produces a new vector whose direction is the same as B but whose magnitude is
mtimes the magnitude of B. Symbolically, |mB|=m|B|.If mis a negative scalar
the direction of mB is opposite to that of the direction of B. In figure 6-1 several
vectors obtained from B by scalar multiplication are exhibited.
5. Vectors are considered as “free vectors”. The term “free vector” is used to mean
the following. Any vector may be moved to a new position in space provided
that in the new position it is parallel to and has the same direction as its original
position. In many of the examples that follow, there are times when a given
vector is moved to a convenient point in space in order to emphasize a special
geometrical or physical concept. See for example figure 6-1.
Vector Addition and Subtraction
Let C =A+B denote the sum of two vectors A and B. To find the vector sum
A+B, slide the origin of the vector B to the terminus point of the vector A, then draw
the line from the origin of Ato the terminus of B to represent C. Alternatively, start
with the vector B and place the origin of the vector A at the terminus point of B to
construct the vector B+A. Adding vectors in this way employs the parallelogram law
for vector addition which is illustrated in the figure 6-2. Note that vector addition is
commutative. That is, using the shifted vectors A and B, as illustrated in the figure
6-2, the commutative law for vector addition A+B=B+A, is illustrated using the
parallelogram illustrated. The addition of vectors can be thought of as connecting
the origin and terminus of directed line segments.
Figure 6-2. Parallelogram law for vector addition
