Apply analytical methods of vector algebra to find resultant vectors and to solve vector equations for unknown vectors.
Interpret physical situations in terms of vector expressions.
Vectors can be added together and multiplied by scalars. Vector addition is associative (
[link] ) and commutative (
[link] ), and vector multiplication by a sum of scalars is distributive (
[link] ). Also, scalar multiplication by a sum of vectors is distributive:
In this equation,
is any number (a scalar). For example, a vector antiparallel to vector
can be expressed simply by multiplying
by the scalar
:
Direction of motion
In a Cartesian coordinate system where
denotes geographic east,
denotes geographic north, and
denotes altitude above sea level, a military convoy advances its position through unknown territory with velocity
. If the convoy had to retreat, in what geographic direction would it be moving?
Solution
The velocity vector has the third component
, which says the convoy is climbing at a rate of 100 m/h through mountainous terrain. At the same time, its velocity is 4.0 km/h to the east and 3.0 km/h to the north, so it moves on the ground in direction
north of east. If the convoy had to retreat, its new velocity vector
would have to be antiparallel to
and be in the form
, where
is a positive number. Thus, the velocity of the retreat would be
. The negative sign of the third component indicates the convoy would be descending. The direction angle of the retreat velocity is
south of west. Therefore, the convoy would be moving on the ground in direction
south of west while descending on its way back.
The generalization of the number zero to vector algebra is called the
null vector , denoted by
. All components of the null vector are zero,
, so the null vector has no length and no direction.
Two vectors
and
are
equal vectors if and only if their difference is the null vector:
This vector equation means we must have simultaneously
,
, and
. Hence, we can write
if and only if the corresponding components of vectors
and
are equal:
Two vectors are equal when their corresponding scalar components are equal.
Resolving vectors into their scalar components (i.e., finding their scalar components) and expressing them analytically in vector component form (given by
[link] ) allows us to use vector algebra to find sums or differences of many vectors
analytically (i.e., without using graphical methods). For example, to find the resultant of two vectors
and
, we simply add them component by component, as follows:
In this way, using
[link] , scalar components of the resultant vector
are the sums of corresponding scalar components of vectors
and
:
