### Video Transcript

Vector π is represented by the
following graph. Which of the following represents a
half π?

In this question, weβre given a
graphical representation of a vector π. We need to determine which of five
given options is the correct graphical representation of a half π. And there are a few different ways
we could answer this question. Weβre going to go through two of
these.

Letβs start by recalling what it
means to multiply a vector by a scalar. And thereβs two different ways of
thinking about multiplying a vector by a scalar. Letβs start with the graphical
interpretation. If we have a vector π― and we
multiply this by a scalar π, then the length of our vector will be multiplied by
π. If π is positive, the direction of
this vector stays exactly the same. However, if π is negative, then we
flip the direction of the vector. And itβs worth noting if π is
equal to zero, weβre multiplying a vector by zero. This gives us the zero vector.

And we can use this to answer our
question. Weβre given a graphical
representation of π, and we want to determine a half times π. So our value of π is one-half,
which is positive. This means we need to find the
vector of one-half length of π, which points in the same direction. And we can do this directly from
the diagram. Thereβs a few different ways we
could do this. One way is to note that the
midpoint of the line segment from zero, zero to two, two is the point one, one. This means it cuts the vector in
half. So, for example, the vector from
zero, zero to one, one is the vector one-half π.

And itβs also worth pointing out
here vectors are only defined by their magnitude and direction. So we can draw these vectors
anywhere in the plane. For example, we couldβve drawn the
vector from one, one to two, two. This would also be the vector
one-half π, since it has the same magnitude and direction as the vector from zero,
zero to one, one. Similarly, we couldβve also drawn
our vector anywhere else in the plane. As long as the vectors have the
same magnitude and direction, the vectors will be equal. And we can see that this is only
true for the graph in option (B). And this allows us to conclude the
answer is option (B).

However, there is a second method
we couldβve used to answer this question. So letβs go through this method
now. Weβll start by clearing some
space. We can then recall a second method
we can use to multiply a vector by a scalar. This is by using the components of
the vector. π times the vector π£ sub one, π£
sub two is equal to the vector π times π£ sub one, π times π£ sub two. In other words, to multiply a
vector by a scalar, we just multiply all of the components of the vector by the
scalar. So another method of determining
the correct graph will be to find the components of vector π and then multiply each
component by one-half. We can then sketch the vector
one-half π. And we can determine the components
of vector π from the graph.

And thereβs two different ways of
doing this. One way is to consider the
horizontal and vertical displacement from the initial point of the vector to its
terminal point. In this case, we move two units to
the right and then two units upwards. This tells us that π is the vector
two, two. However, a second way of doing this
is to use the fact that we know the coordinates of the initial point and terminal
points of vector π. We can just recall that each
component of a vector is equal to the difference in each coordinate of its terminal
point and its initial point. So π is the vector two minus zero,
two minus zero, which simplifies to give us the vector two, two.

We can now use this to determine
one-half times vector π. We need to multiply each component
of vector π by one-half. This gives us the vector one-half
times two, one-half times two. And one-half times two is one. So one-half π is the vector one,
one. And now since all five of the given
options start at the origin, letβs sketch the vector one-half π starting at the
origin. Its π₯-component is one, and its
π¦-component is also one. So we travel one unit to the right
and one unit up to the point with coordinates one, one. And this also confirms that our
answer is option (B). Therefore, if π is represented by
the given graph, we were able to show that only option (B) represents the vector
one-half π.