Consider vectors and
in Figure 4.29. In component form these are written as
and
. How can we calculate the scalar product
?
The scalar product will tell us how much vector will grow vector
, and to determine this we want to identify how much the vectors interact. One method is to consider how much the horizontal and vertical components of the vectors interact, as illustrated in Figure 4.30. There are four possible combinations to consider: horizontal to horizontal, horizontal to vertical, vertical to horizontal, and vertical to vertical.
Horizontal components do not interact with vertical components (and vice versa) because they are independent of each other, so and
, and they do not contribute to the value of scalar product. Horizontal components interact with each other, and vertical components interact with each other, so
and
both contribute to the value of
.
The expression is a measure of how much the scalar quantity
grows the scalar quantity
, so it is equal to
, and similarly
is equal to
. The scalar product is a combination of these, so
For example, if and
, then
If and
, or
and
in column notation, then
Suppose that ,
and
. Find the following.
OpenLearn - Part 2: Chapter 4 Applications of vectors Except for third party materials and otherwise, this content is made available under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 Licence, full copyright detail can be found in the acknowledgements section. Please see full copyright statement for details.