### Video Transcript

Express the vector π with
components negative five over two and negative 19 using the unit vectors π’ and
π£.

So weβre given these two unit
vectors π’ and π£. These are sometimes written with
circumflex accents on top which look like hats. This emphasises the fact they are
unit vectors, that is vectors whose magnitude is one. Of course, there are many unit
vectors. There are unit vectors pointing in
any direction you like. π’ and π£ are particular unit
vectors; π’ is the unit vector pointing in the π₯-direction, and π£ is the unit
vector pointing in the π¦-direction. π’ has components one, zero and π£
has components zero, one.

Any two-dimensional vector π can
be written in terms of π’ and π£. The vector π is equal to the
π₯-component of π£ times π’ plus the π¦-component of π£ times π£. So the vector π is equal to the
π₯-component of π, negative five over two, times π’ plus the π¦-component of π,
negative 19, times π£.

Writing the plus negative 19π£ as
minus 19π£, we get our final answer: π is equal to negative five over two π’ minus
19π£.

We can check this using the
components of π’ and π£ and what we know about the scale of multiplication and
subtraction of vectors. Negative five over two π’ minus
19π£ is equal to negative five over two times the vector with components one, zero
minus 19 times the vector with components zero, one. Here weβre using the component
forms of π’ and π£. And of course, when multiplying a
vector by a scaler, we just multiply the components of that vector by the
scaler. So we get the vector with
components negative five over two, zero minus the vector with components zero,
19. And subtracting one vector from
another means subtracting the components of that vector from the other.

So we get the vector with
components negative five over two, negative 19. And as desired, this is the vector
π.