# Video: Multiplying Vectors by Scalars and Adding and Subtracting Vectors Component-Wise

Given that vector 𝐴 = (−2, 2), vector 𝐵 = (5, 2), and vector 𝐶 = (−3, −2), find − vector 𝐴 + vector 𝐵 − vector 𝐶.

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### Video Transcript

Given that vector 𝐴 equals negative two, two; vector 𝐵 equals five, two; and vector 𝐶 equals negative three, negative two, find negative vector 𝐴 plus vector 𝐵 minus vector 𝐶.

Whenever we’re looking to solve a problem that’s adding and subtracting vectors, what we want to do is actually split it into its component parts. So we’re gonna begin by looking at the 𝑥 components. So our first term is negative negative two — and this is because it’s negative vector 𝐴 — plus five, because that’s the 𝑥-component of vector 𝐵, and then minus negative three. And that’s because this is the 𝑥-component of vector 𝐶. And again, we’ve got minus and negative.

Okay, now we can move on to the 𝑦-components of our vectors. So we’re gonna start with negative two — and that’s because it’s minus vector 𝐴 — then plus two — because this is the 𝑦-component of vector 𝐵. And then, finally, we have the 𝑦-component of vector 𝐶. So we’ve got minus negative two. And again, it’s negative negative because we’re subtracting the final vector.

Okay, great! So now what we want to do is actually calculate each of our components. So for our 𝑥-component, we have two plus five plus three. And that’s because I’ve just tidied it up because we had minus and negative, which makes a positive, four minus negative two and minus negative three. And then for our 𝑦-component, we have negative two plus two plus two. And again, we got this because you’re subtracting a negative. So therefore, it turns positive.

So therefore, we can say that, given that vector 𝐴 equals negative two, two; vector 𝐵 equals five, two; and vector 𝐶 equals negative three, negative two, then minus vector 𝐴 plus vector 𝐵 minus vector 𝐶 is equal to 10, two, where our 𝑥-component is 10 and our 𝑦-component is two.