The vector 𝚨 can be written in the form 𝑎 subscript 𝑥 𝐢 hat plus 𝑎 subscript 𝑦 𝐣 hat. What is the value of 𝑎 subscript 𝑥 and what is the value of 𝑎 subscript 𝑦?
Let’s first define 𝐢 hat and 𝐣 hat. These are unit vectors. So if we define one square in our graph to be one unit, then 𝐢 hat is one square in the horizontal direction and 𝐣 hat is one square in the vertical direction. 𝑎 subscript 𝑥 is then the number we multiply by the unit vector to make the horizontal component of the vector 𝚨.
The tail of vector 𝚨 is on the vertical axis. So we just need to count the distance from the axis to the tip of vector 𝚨 to get the horizontal component. So starting from the axis, that’s one, two, three, four, five. And note that when we go from the tail to the tip of 𝚨, we’re counting to the left in the horizontal direction, whereas the unit vector 𝐢 went upwards to the right, which makes it negative five. So the answer to “What is the value of 𝑎 subscript 𝑥?” is negative five.
Next, we need to find the value of 𝑎 subscript 𝑦. So we’re doing the same thing in the vertical direction. The tail of vector 𝚨 is on the horizontal axis. So starting from the axis towards the tip of vector 𝚨, we have one, two, three, four. And that was in the downwards direction, whereas the unit vector 𝐣 hat counts upwards, making this negative four. So the answer to “What is the value of 𝑎 subscript 𝑦?” is negative four.