The diagram shows two vectors 𝐀 and 𝐁. Work out 𝐀 minus 𝐁 in component form.
One way to approach this is to first calculate 𝐀 and 𝐁 in component form, which means in terms of 𝐢 hat, the unit vector in the horizontal direction, and 𝐣 hat, the unit vector in the vertical direction. For 𝐀, we have negative one, two, three for the horizontal component. And for the vertical component, we have positive one, two, three, four, five. So we can write 𝐀 in component form as negative three 𝐢 plus five 𝐣. For 𝐁, we have positive one, two, three, four for the horizontal component and positive one, two for the vertical component. So we can write 𝐁 as four 𝐢 plus two 𝐣.
Now we can calculate 𝐀 minus 𝐁 in component form by subtracting the components of 𝐁 separately from the components of 𝐀. So for the horizontal component, we have negative three minus four, which is equal to negative seven. And for the vertical component, we have positive five minus two, which is equal to three. Now we can draw this resultant vector on our diagram by counting up the horizontal component as negative one, two, three, four, five, six, seven and the vertical component as positive one, two, three. So this is our resultant vector 𝐀 minus 𝐁.
We can check this is right using the graphical tip-to-tail method. Starting from the tip of vector 𝐀, we’ll draw in negative 𝐁, which will have horizontal component of negative four and vertical component of negative two. The resultant vector, 𝐀 minus 𝐁, then goes from the tail of vector 𝐀 to the tip of vector negative 𝐁. And we can see that the vector we already found does that. Therefore, 𝐀 minus 𝐁 is equal to negative seven 𝐢 plus three 𝐣.