### Video Transcript

The diagram shows three vectors:
π, π, and π. Which of the following expressions
gives π in terms of π and π? Is it (A) π equals π plus π? (B) π equals π minus π. (C) π equals π minus π. (D) π equals two π minus π. Or (E) π equals two π minus
π.

To answer this question, we need to
work out how to either add or subtract some combination of vectors π and π to
produce vector π. This is a good opportunity for us
to practice adding or subtracting vectors algebraically.

So letβs first recall we can
represent a vector as a sum of multiples of unit vectors along the π₯- and
π¦-axes. To see what we mean, letβs look at
this given vector π. Note that it starts at the origin
and points left one unit and down six units. Now, as usual, weβll define the
rightward direction as positive along the π₯-axis and upward as positive along the
π¦-axis. So, since the vector π points left
one unit, we say that it has a horizontal component of negative one unit. Likewise, we say that π has a
vertical component of negative six units, since weβve defined downward as
negative.

At this point, we should recall
that the unit vector along the π₯-axis is called π’ hat and that it represents a
vector of one unit length in the positive π₯-direction. Similarly, the unit vector along
the π¦-axis is called π£ hat, and it represents a vector of one unit length in the
positive π¦-direction. So to represent the vector π
algebraically, we say that π equals negative one times π’ hat plus negative six
times π£ hat or just negative one π’ hat minus six π£ hat.

Next, letβs look at vector π. It also starts at the origin, and
then points two units upward and five units to the left. Thus, we say that vector π equals
negative five π’ hat plus two π£ hat.

Okay, now that weβve defined
vectors π and π, letβs start to think about how we could combine them to create
vector π. Recall that when we add or subtract
vectors algebraically, we simply add or subtract their like components: π’ hat with
π’ hat and π£ hat with π£ hat. For example, letβs see what happens
when we add π and π. Adding their horizontal components,
negative one π’ hat plus negative five π’ hat gives negative six π’ hat. And adding their vertical
components, negative six π£ hat plus two π£ hat gives negative four π£ hat.

So, is this vector π plus π equal
to vector π, like answer option (A) suggests? To find out, letβs use the diagram
to determine the horizontal and vertical components of π. Note that the tail of the vector is
located at this point here, rather than the origin. So, starting here, π points four
units to the right and eight units downward. This does not correspond to the
vector π plus π that we just found. So we can eliminate option (A). Instead, vector π is represented
algebraically as four π’ hat minus eight π£ hat.

So now, rather than going through
and working out all of the remaining answer options, letβs separately consider the
π₯- and π¦-components of all three of these vectors and think about to combine like
components of π and π to give the respective component of π. We can start with the vertical
components.

Think: how can we combine negative
six and two to make negative eight? Well, we know that negative six
minus two gives negative eight. So thatβs a good hint that vector
π minus vector π gives vector π. Now, checking the horizontal
components, subtracting vector π from π gives negative one minus negative five,
which equals positive four, exactly equal to the horizontal component of vector
π. This confirms that vector π minus
vector π equals vector π. So we know that answer option (B)
is correct. This expression correctly gives
vector π in terms of vectors π and π.