Video: Finding the Norm of the Difference of Two given Vectors in Three Dimensions

If vector 𝐴 = 4𝑖 + 4𝑗 βˆ’ 5π‘˜ and vector 𝐡 = 3𝑖 βˆ’ π‘˜, determine β€–vector 𝐴 βˆ’ vector 𝐡‖.

02:33

Video Transcript

If vector 𝐴 is equal to four 𝑖 plus four 𝑗 minus five π‘˜ and vector 𝐡 is equal to three 𝑖 minus π‘˜, determine the magnitude or modulus of vector 𝐴 minus vector 𝐡.

Well in this problem, the first thing we want to do is vector 𝐴 minus vector 𝐡. And when we do this, so when we subtract vector 𝐡 from vector 𝐴, what we do is we deal with each individual components separately. So first of all, we have four minus three 𝑖. And then we add on four minus zero 𝑗. And that’s because you get four from four 𝑗 in vector 𝐴 and then zero because there is no 𝑗 component to vector B. And then finally, we add on negative five minus negative one π‘˜. And that’s because you get negative five in vector 𝐴 and then negative π‘˜, so negative one, from vector 𝐡.

So we can say that vector 𝐴 minus vector 𝐡 is gonna be equal to 𝑖 as our first term. And that’s because we had four minus three which is just one. So it’s just 𝑖 plus four 𝑗 and then minus four π‘˜. And that’s because we had negative five minus negative one. Well, minus a negative is an add. So you get negative five add one which is negative four.

So now what we need to do is we need to find the magnitude or the modulus of vector 𝐴 minus vector 𝐡. Well, what is the magnitude? Well, if you want to find the magnitude of a vector, then it’s equal to the square root of π‘₯ squared plus 𝑦 squared plus z squared where these are the coefficients of 𝑖, 𝑗, and π‘˜. So, therefore, the magnitude of vector 𝐴 minus vector 𝐡 is gonna be equal to one squared plus four squared plus negative four squared. And that’s because these were the coefficients of 𝑖, 𝑗, and π‘˜, respectively.

So, therefore, this is gonna be equal to the square root of one plus 16 plus 16 which is gonna be equal to root 33. So, therefore, we can say that if vector 𝐴 is equal to four 𝑖 plus four 𝑗 minus five π‘˜ and vector 𝐡 is equal to three 𝑖 minus π‘˜, then the magnitude of vector 𝐴 minus vector 𝐡 is going to equal to root 33.

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