Write 𝐀 in component form.
Now, the first thing to be aware of here is that 𝐀 marked on the diagram is written in bold, whereas when we handwrite it, we put this little half arrow over the top. These are both just ways of indicating that 𝐀 is a vector, meaning it has both magnitude and direction. Now, what we mean by writing 𝐀 in component form is we want to write it in the form 𝐀 is equal to 𝑎 sub 𝑥 𝐢 hat plus 𝑎 sub 𝑦 𝐣 hat. Here, 𝐢 hat and 𝐣 hat are unit vectors. Since our vector 𝐀 is drawn on a grid, we can define one grid square to be one unit. And then 𝐢 hat is one unit in the horizontal direction, and 𝐣 hat is one unit in the vertical direction.
Since 𝐢 hat and 𝐣 hat are providing the direction of our vector, we then have 𝑎 sub 𝑥 and 𝑎 sub 𝑦 to provide the magnitude of the vector, where 𝑎 sub 𝑥 is the magnitude in the horizontal direction and 𝑎 sub 𝑦 is the magnitude in the vertical direction. So let’s first find 𝑎 sub 𝑥. We do this by starting at the tail of vector 𝐀, which is at the origin, and then counting horizontally one, two, three, four, five, six until we’re in line with the tip of the vector. That was six grid squares and in the same direction as the unit vector 𝐢 hat towards the right of the screen. Therefore, 𝑎 sub 𝑥 is equal to six.
Now we can do the same thing to find 𝑎 sub 𝑦. This time we can start from the end of our horizontal component and then count grid squares vertically until we reach the tip of 𝑎. So that’s one, two, three units. And note that this time we were going down, which is the opposite direction to the unit vector 𝐣 hat, which is one unit upwards towards the top of the screen. Therefore, the vertical component 𝑎 sub 𝑦 is equal to negative three. So we can now insert these numbers, and we have 𝐀 is equal to six 𝐢 hat minus three 𝐣 hat. And now we have the vector 𝐀 written in component form.