### Video Transcript

The given figure shows a vector in a plane. Express this vector in terms of the unit vectors π and π.

π and π are both unit vectors. So the magnitude of π and the magnitude of π are both one. But the vector quantities have both magnitude and direction. So we need to know also which directions π and π point in.

By definition, π is the unit vector in the π₯-direction. We can draw it wherever weβd like to in the plane. Iβve chosen the second quadrant. And by definition, π is the unit vector in the π¦-direction.

So hereβs one copy of π in the third quadrant. We have a vector. Letβs call it π£. And our task is to write this vector π£ in terms of π and π. What does this mean?

Recall that we can add two vectors together by placing them nose-to-tail. If I add π to itself, I get the vector pointing in the π₯-direction with a magnitude of two. This is π plus π or two π. Similarly, if I put a three copies of π together nose-to-tail, I get three π.

Now weβre ready to tackle our vector π£. We can get from the initial point of π£ to the terminal point of π£ by first going two units right and then going 10 units up. The orange and purple vectors are nose-to-tail or, more formally, the terminal point of the orange vector is the initial point of the purple vector.

And so our green vector π£ is equal to the orange vector plus the purple vector. We recognize the orange vector. It has a magnitude of two and is pointing in the π₯-direction. And so it is two π. How about the purple vector which points in the π¦-direction and has a magnitude 10? Well, this is just 10π.

Our vector π£, therefore, is two π plus 10π. Weβve succeeded in writing this vector in terms of the unit vectors π and π.