Video Transcript
The diagram shows three vectors 𝐀, 𝐁, and 𝐂. The grid squares in the diagram have a side length of one. What is 𝐀 plus 𝐁 plus 𝐂 in component form?
What we’re looking for in this question is to express the resultant vector of 𝐀 plus 𝐁 plus 𝐂 in the form 𝑎 sub 𝑥 𝐢 hat plus 𝑎 sub 𝑦 𝐣 hat, where 𝑎 sub 𝑥 is the number of units in the horizontal direction and 𝑎 sub 𝑦 is the number of units in the vertical direction. 𝐢 hat and 𝐣 hat are the unit vectors in the horizontal and vertical directions, respectively. And the question tells us they’re equivalent to one square on the grid.
To add the vectors together, we can use the tip-to-tail method. So we start from the tip of vector 𝐀, which is up here, and we’re going to slide vector 𝐁 over so that its tail touches the tip of 𝐀. When we’re doing this, we have to make sure our new vector is the same as the original in both magnitude and direction. So in the vertical direction, which is conventionally counted upwards, we have positive two. And then the horizontal direction is conventionally counted to the right, so that’s negative six in the horizontal direction. And we have our new vector 𝐁. We then do the same thing with vector 𝐂, sliding it over so that its tail touches the tip of vector 𝐁. And then the resultant vector from the tail of vector 𝐀 to the tip of vector 𝐂 is the sum of 𝐀 plus 𝐁 plus 𝐂.
Now we need to write this in component form. So let’s look at the two components of our resultant vector 𝐀 plus 𝐁 plus 𝐂. We have negative two in the horizontal direction. So that gives us 𝑎 sub 𝑥. And then positive one in the vertical direction gives us 𝑎 sub 𝑦 equals one. So the resultant vector 𝐀 plus 𝐁 plus 𝐂 is negative two 𝐢 hat plus one 𝐣 hat, and that’s our resultant vector written in component form.
