Video Transcript
Write 𝐀 in component form.
Here, we see this vector 𝐀 drawn
on a grid. And we can see the vector starts at
the origin of a coordinate frame. Let’s call the horizontal axis the
𝑥-axis and the vertical one the 𝑦. Now, when we go to write this
vector 𝐀 in component form, that means we’ll write it in terms of an 𝑥- and a
𝑦-component, also called a horizontal and vertical component. If we call the 𝑥-component of
vector 𝐀 𝐴 sub 𝑥 and the 𝑦-component 𝐴 sub 𝑦, then we can multiply each one of
these components by the appropriate unit vector. The unit vector for the 𝑥- or
horizontal direction is 𝐢 hat, and the unit vector for the vertical or 𝑦-direction
is 𝐣 hat. By themselves, the 𝑥- and
𝑦-components of vector 𝐀 are not vectors; they’re 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, we’ll get the vector 𝐀. Expressing 𝐀 this way is known as
writing it in component form. So then, what are 𝐴 sub 𝑥 and 𝐴
sub 𝑦? To figure that out, we’ll need to
look at our grid. Starting with 𝐴 sub 𝑥, that’s
equal to the horizontal component of this vector 𝐀. In other words, if we project this
vector perpendicularly onto the 𝑥-axis, then the length of that line segment, this
length here, is 𝐴 sub 𝑥. 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 𝑥-values. So even though this horizontal
orange line is three units long, we say that the 𝑥-component of 𝐀 is negative
three. This is because the projection of
vector 𝐀 onto the horizontal axis goes negative three units in the
𝑥-direction.
To find the vertical component of
𝐀, we’ll follow a similar process. Once again, we project vector 𝐀
perpendicularly, this time onto the vertical axis. And it’s the length of this line
that tells us the vertical or 𝑦-component of 𝐀. We see that this is one, two units
long and that this is in the positive 𝑦-direction. 𝐴 sub 𝑦 then is equal to positive
two. And now we can write out 𝐀 in its
component form. Vector 𝐀 is equal to negative
three times the 𝐢 hat unit vector plus two times the 𝐣 hat unit vector.
