Activity 4.20 illustrates two important properties of the scalar product. First, if two non-zero vectors are perpendicular, then their scalar product is zero. This is because if and
are perpendicular, then
This also works the other way, so that if the scalar product of two non-zero vectors is zero, then the vectors are perpendicular. This is because if and
are non-zero vectors, then the only way that
can be equal to zero is if
. This implies that
.
The second property is that the scalar product of a vector with itself is equal to the square of the magnitude of the vector. This is because if is any
non-zero vector, then the angle between and itself is 0°, so
These, and other, properties of the scalar product in the following list can all be proved using the definition of the scalar product in a similar way.
The following properties hold for all vectors ,
and
, and every scalar
.
If and
are non-zero and perpendicular, then
and
.
We can use these properties to simplify expressions containing scalar products of vectors.
Expand and simplify the expression , where
and
are vectors.
Expand the brackets by using property 4:
Simplify by using property 3, so:
Using property 2, simplify further to get:
Expand and simplify the expression , where
and
are vectors.
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