Consider two vectors 𝐀 and 𝐁, where 𝐀 equals eight 𝐢 hat minus four 𝐣 hat and 𝐁 equals negative two 𝐢 hat plus five 𝐣 hat. Calculate 𝐀 minus 𝐁.
So in this question, we’re given two vectors 𝐀 and 𝐁 in component form and we’re asked to calculate 𝐀 minus 𝐁. So that’s the result when we subtract the vector 𝐁 from the vector 𝐀. Looking at our two vectors 𝐀 and 𝐁, then if we recall that 𝐢 hat is the unit vector in the 𝑥-direction and 𝐣 hat is the unit vector in the 𝑦-direction, then we can see that each of our vectors has an 𝑥-component and a 𝑦-component.
We should recall that if we want to subtract one vector from another, then we can do this by subtracting the 𝑥- and the 𝑦-components of those vectors separately. With this in mind, we can calculate 𝐀 minus 𝐁. If we first subtract the 𝑥-components, then we see that we have the 𝑥-component of 𝐀, which is eight, minus the 𝑥-component of 𝐁, which is negative two. So our 𝑥-component of 𝐀 minus 𝐁 is equal to eight minus negative two. And since it’s an 𝑥-component, we multiply this by the unit vector 𝐢 hat.
Then, if we subtract the 𝑦-components, we have the 𝑦-component of 𝐀, which is negative four, minus the 𝑦-component of 𝐁, which is five. So our 𝑦-component of 𝐀 minus 𝐁 is equal to negative four minus five. And this 𝑦-component gets multiplied by the unit vector 𝐣 hat.
Then, evaluating our 𝑥- and 𝑦-components, we find that the 𝑥-component eight minus negative two gives us 10 and the 𝑦-component negative four minus five gives us negative nine. And so we found our answer to the question that when we calculate 𝐀 minus 𝐁, our result is 10𝐢 hat minus nine 𝐣 hat.