### Video Transcript

The diagram shows two vectors, π
and π. Each of the grid squares in the
diagram has a side length of one. Calculate π dot π.

We see over in our diagram our two
vectors, π and π, and that theyβre laid out on a grid spacing. Weβre told that each one of these
grid squares has a side length of one. Weβre not told the units of these
lengths, but simply that the side lengths can be represented by one single unit,
whatever our unit is. Knowing this, we want to calculate
the scalar product of π and π.

Now, we can recall that a scalar
product involves combining two vectors. So, weβre off to a good start there
because π and π are vectors. And we can recall further that,
mathematically, the scalar product of two general vectors, π and π, is equal to
the product of their π₯-components plus the product of their π¦-components. Now, for our two specific vectors,
also called π and π, we donβt yet know their π₯- and π¦-components, but we can use
this grid to find out.

We can start by laying down
coordinate axes on this grid. Letβs say that for our origin, we
pick the location where the tails of vectors π and π overlap. So, weβll say that this is our
π₯-axis, and this is our π¦. Relative to these axes, we can
define the π₯- and π¦-components of our two vectors. Just as a side note, we could pick
any orientation for our π₯- and π¦-axes so long as theyβre perpendicular to one
another and quantified vectors π and π that way. And our answer would come out the
same.

Using these specific π₯- and
π¦-axes though, letβs write out the components of vector π. We can see that along the π₯-axis,
vector π extends one, two, three units. So, that means vector π is equal
to three π’, three units in the π₯-direction, plus some amount in the
π¦-direction. Starting again at the origin, we
count up one, two, three units and see that this is the vertical extent of vector
π. Therefore, we can write vector π
as three π’ plus three π£. And now, weβll do the same thing
for vector π. The π₯-component of vector π is
equal to one, two, three, four, five, six units and its π¦-component is equal to one
unit. And we can write that as one π£ or
simply π£ so that vector π overall is equal to six π’ plus π£.

Now that we know the components of
our two vectors, we can use this relationship to solve for their scalar product. π dot π is equal to the
π₯-component of vector π, we see that π₯-component is three, multiplied by the
π₯-component of vector π. And we see that π₯-component is
six. So, we have three times six. And to that, we add the
π¦-component of vector π. That π¦-component is three
multiplied by the π¦-component of vector π. And that π¦-component as we saw is
one. So, π dot π is equal to three
times six plus three times one, and that is equal to 18 plus three or 21. This is π dot π, also called the
scalar or dot product of π and π.