Consider two vectors: 𝐀 and 𝐁. 𝐀 equals negative seven 𝐢 hat minus seven 𝐣 hat and 𝐁 equals negative six 𝐢 hat minus two 𝐣 hat. Calculate 𝐀 plus 𝐁.
This question gives us two vectors in component form, and it asks us to calculate their sum. Our first vector is 𝐀, which equals negative seven 𝐢 hat minus seven 𝐣 hat. Since 𝐢 hat is the unit vector in the 𝑥-direction and 𝐣 hat is the unit vector in the 𝑦-direction, this means that 𝐀 extends negative seven units in the 𝑥-direction and negative seven units in the 𝑦-direction. So, vector 𝐀 looks like this. Our second vector is 𝐁, which equals negative six 𝐢 hat minus two 𝐣 hat. This means that vector 𝐁 extends negative six units in the 𝑥-direction and negative two units in the 𝑦-direction. So it looks like this.
Now we need to add these two vectors together. The question gives us these vectors in component form. And in this case, the simplest way to add these two vectors is to do it algebraically. To do this, we add together the 𝑥-components and the 𝑦-components of the two vectors separately. The result of this, the sum of these two vectors, is known as their resultant. So let’s take our two vectors 𝐀 and 𝐁 and add together their 𝑥- and 𝑦-components.
If we add together the 𝑥-components to get the 𝑥-component of our resultant vector, we have negative seven plus negative six. Since this is the 𝑥-component, we multiply this by 𝐢 hat. Then if we add the 𝑦-components, we have negative seven plus negative two. And this gets multiplied by 𝐣 hat. The final step is to evaluate these sums for the 𝑥-component and the 𝑦-component. For the 𝑥-component, adding together negative seven and negative six gives us a result of negative 13. And for the 𝑦-component, adding negative seven and negative two gives us negative nine. And so we find that the sum or the resultant of the vectors 𝐀 and 𝐁 is equal to negative 13𝐢 hat minus nine 𝐣 hat.