Two Dimensional Vectors (Level 8) In this video we are going to cover properties
of vector operations. It turns out that vector addition and scalar multiplication share many
properties of ordinary arithmetic, the following are properties associated with vector operations:
Let u, v and w be vectors in a plane, and let c and d be scalars. Then we can express
the following properties as follows: Vector u plus vector v equals vector v plus
vector u in other words vector addition is commutative, this is known as the commutative
property of vector addition, this is similar to the commutative property of addition for
real numbers which you learned way back in your algebra I course. The only difference
is that we are now dealing with vectors as oppose to real numbers. This property can
be verified either geometrically or algebraically. For a geometric proof we simply use the parallelogram
law to illustrate this property, notice that regardless of the order in which we add vectors
v and u we obtain the same resultant vector. Alternatively for an algebraic proof we essentially
need to compute the left side of the vector expression and then do some basic arithmetic
on the result to show that we can make the left side match the right side of the vector
expression. Similar to the way you proved trigonometric identities.
For this property we simply start with the left hand side of the expression and break
the vectors into their component forms as follows. Next we go ahead and add the vectors
component wise, for the next step we are going to use the commutative property of addition
for real numbers and rearrange the order in which we add the vectors since addition of
real numbers is commutative for example 2 + 3 yields the same answer as 3+2, notice
that we are not rearranging the vectors but rather the components of the vectors which
are real numbers this is why we are allowed to use the commutative property of real numbers.
Next we notice that this new arrangement is nothing more than adding the components of
vector v and the components of vector u, lastly these rearranged components represent vector
v and vector u respectively and this completes the proof. We can denote the ending of a proof
by using the letters Q.E.D which means quod erat demonstrandum which is latin for “that
which was to be demonstrated” or "which had to be proven" or "what was to be demonstrated"
or "the very thing it was required to have shown." You can also use a tombstone mark which
is small rectangle that means “end of proof”. This is the typical method you want to follow
as you proof various properties throughout your multivariable calculus course.
Let’s take a look at our second property. The sum of vector u and vector v plus vector
w is equal to vector u plus the sum of vector v and vector w. This is known as the associative
property for vector addition. Again this is similar to the associative property of addition
for real numbers. We are free to add vectors in any order without affecting the resulting
vector. We can geometrically proof this property by using the following diagram: notice that
the sum of vector u and vector v plus vector w yields the same resultant vector when adding
vector u plus the sum of vector v and vector w. Alternatively we can prove it algebraically
as follows:
We break apart each vector into its component form, then we go ahead and add vector u and vector
v component wise this will yield the following vector, next we go ahead and add these two
vectors component wise, this results in the following resultant vector. Now the next step
requires us to use the associative property of real numbers to group the real numbers
in a different order as follows. Then it is just a matter of working backwards and rewriting
the expressions until we match the target vector expression, as follows. The rest of
the vector properties can be proven in a similar fashion, many of the vector properties will
depend on similar properties of real numbers to carry out the proof. Ok let’s move along
to the next property.
Vector u plus the zero vector is equal to vector u, this is known as the additive identity
property for vectors. This property also resembles the additive identity property for real numbers.
Our next property is, vector u plus the negative of vector u is equal to the zero vector, this
is known as the additive inverse property for vectors.
Ok, notice that these 4 properties involved only vectors, now let’s take a look at vector
properties that involve both scalars and vectors.
Scalar c times scalar d times vector u is equivalent to the product of scalar c and
d times vector u. Notice that the scalars are following the associative property of
multiplication. In a later video we will learn about vector multiplication for now keep in
mind that this property only applies to scalars, I repeat do not confuse it with vector multiplication
which is a totally different concept from scalar multiplication.
The next property is, the scalar quantity c + d times vector u is equal to scalar c
times vector u plus scalar d times vector u. In this property the vector is following
the distributive property of real numbers. Once again notice that this property is applied
to scalar quantities and not vectors. We will define vector multiplication in a much later
video. Similar to the previous property, scalar c
times the sum of vector u and v is equal to scalar c times vector u plus scalar c times
vector v. This is essentially another form of the distributive property for scalars.
Notice that it is the scalar that is being distributed to each of the vectors.
Next, scalar 1 times vector u is equal to vector u, this is similar to the Identity
property of multiplication also known as the multiplication property of one, for real numbers.
Lastly we have, scalar 0 times vector u is equal to the 0 vector. This is similar to
the zero product property of multiplication which is also known as the zero product property
for real numbers. Once again these 9 properties of vectors can
be verified either geometrically or algebraically. In general any set of vectors (with an accompanying
set of scalars) that satisfies properties 1 through 8 is a vector space. The eight properties
are referred to as the vector space axioms. In essence the set of vectors in the plane
with a set of real numbers or scalars forms a vector space. Vector spaces will be discussed
in great detail in a first course in linear algebra which is usually taken during or immediately
after this course. Lastly let’s end the video by taking a look
at one final vector property, a property that involves the length or magnitude
of a vector along with a scalar is the length of a scalar multiple.
The magnitude of a scalar times a vector is equal to the absolute value of the scalar
times the magnitude of the vector. This can easily be proven.
We start by denoting vector v in component form as follows, and then we go ahead and
apply the distributive property by distributing scalar c to each of the components as follows.
Next we find the magnitude of this vector by applying the Pythagorean theorem we essentially
need to square both the x component and y component of the vector as follows, then we
go ahead and factor out the c squared term, next we break apart the product in the radicand.
Now it is just a matter of simplifying the expressions, recall that the square root
of a quantity squared is by definition the absolute value of that quantity so we can
rewrite this radical as the absolute value of c. Lastly the radical expression on the
right is essentially the definition of the magnitude of vector v, so we have shown that
the left hind side is equal to the right hand side and this ends the proof.
Alright in our next video we will go over and cover unit vectors.