Question Video: Subtracting Two Vectors Given in Component Form Physics

Consider two vectors 𝐀 and 𝐁, where 𝐀 = 8𝐒 hat βˆ’ 4𝐣 hat and 𝐁 = βˆ’2𝐒 hat + 5𝐣 hat. Calculate 𝐀 βˆ’ 𝐁.


Video Transcript

Consider two vectors 𝐀 and 𝐁, where 𝐀 equals eight 𝐒 hat minus four 𝐣 hat and 𝐁 equals negative two 𝐒 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 𝐒 hat is the unit vector in the π‘₯-direction and 𝐣 hat is the unit vector in the 𝑦-direction, then we can see that each of our vectors has an π‘₯-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 π‘₯- and the 𝑦-components of those vectors separately. With this in mind, we can calculate 𝐀 minus 𝐁. If we first subtract the π‘₯-components, then we see that we have the π‘₯-component of 𝐀, which is eight, minus the π‘₯-component of 𝐁, which is negative two. So our π‘₯-component of 𝐀 minus 𝐁 is equal to eight minus negative two. And since it’s an π‘₯-component, we multiply this by the unit vector 𝐒 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 π‘₯- and 𝑦-components, we find that the π‘₯-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 10𝐒 hat minus nine 𝐣 hat.

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