### Video Transcript

Write π in component form.

Here, we see this vector π drawn
on a grid. And we can see the vector starts at
the origin of a coordinate frame. Letβs call the horizontal axis the
π₯-axis and the vertical one the π¦. Now, when we go to write this
vector π in component form, that means weβll write it in terms of an π₯- and a
π¦-component, also called a horizontal and vertical component. If we call the π₯-component of
vector π π΄ sub π₯ and the π¦-component π΄ sub π¦, then we can multiply each one of
these components by the appropriate unit vector. The unit vector for the π₯- or
horizontal direction is π’ hat, and the unit vector for the vertical or π¦-direction
is π£ hat. By themselves, the π₯- and
π¦-components of vector π are not vectors; theyβre scalar quantities. But when we multiply these scalars
by a vector, the unit vectors, the result is a vector.

Finally, adding these vector
components together, weβll get the vector π. Expressing π this way is known as
writing it in component form. So then, what are π΄ sub π₯ and π΄
sub π¦? To figure that out, weβll need to
look at our grid. Starting with π΄ sub π₯, thatβs
equal to the horizontal component of this vector π. In other words, if we project this
vector perpendicularly onto the π₯-axis, then the length of that line segment, this
length here, is π΄ sub π₯. In terms of the units of this grid,
that length is one, two, three units long. And notice that we moved to the
left of the origin, that is, into negative π₯-values. So even though this horizontal
orange line is three units long, we say that the π₯-component of π is negative
three. This is because the projection of
vector π onto the horizontal axis goes negative three units in the
π₯-direction.

To find the vertical component of
π, weβll follow a similar process. Once again, we project vector π
perpendicularly, this time onto the vertical axis. And itβs the length of this line
that tells us the vertical or π¦-component of π. We see that this is one, two units
long and that this is in the positive π¦-direction. π΄ sub π¦ then is equal to positive
two. And now we can write out π in its
component form. Vector π is equal to negative
three times the π’ hat unit vector plus two times the π£ hat unit vector.