The diagram shows two vectors 𝐀 and 𝐁. The grid squares in the diagram have a side length of one. What is 𝐀 plus 𝐁 in component form?
Now, there are two methods we can use to approach this question. So let’s first do it the graphical way. We’ll do this using the tip-to-tail method, where we start from the tip of vector 𝐀 and then we’re going to slide vector 𝐁 over so that the tail of vector 𝐁 touches the tip of vector 𝐀.
Now, when moving a vector, we have to be careful to ensure it stays the same size. So let’s first count the grid squares in vector 𝐁. We’re going one, two, three vertically and then negative one, two, three horizontally. Now, if we redraw vector 𝐁 starting from the tip of vector 𝐀, we end up with this new vector 𝐁. We can then draw in our resultant vector, which goes from the tail of vector 𝐀 to the tip of the new vector 𝐁. And this is our resultant vector 𝐀 plus 𝐁.
To write this resultant vector in component form, we then count grid squares of the resultant vector. We have negative one in the horizontal direction, so that gives us negative one 𝐢 hat, and then negative one, two in the vertical direction, which gives us minus two 𝐣 hat. And so 𝐀 plus 𝐁 in component form is negative one 𝐢 hat minus two 𝐣 hat.
Now, we could also have approached this question by summing components. Vector 𝐀 in component form is one, two in the horizontal direction, which gives us two 𝐢 hat, and negative one, two, three, four, five in the vertical direction, giving us minus five 𝐣 hat. Vector 𝐁 is negative one, two, three horizontally, so that’s negative three 𝐢 hat, and positive one, two, three vertically, so we have plus three 𝐣 hat.
Now, we add these together by summing the components individually. So we have two plus negative three, which gives us negative one 𝐢 hat, and then negative five plus three, which gives us negative two 𝐣 hat.
So we can see that both methods give us the same answer that 𝐀 plus 𝐁 is equal to negative one 𝐢 hat minus two 𝐣 hat.