In Exercises 27–30, letv = i - 5j and w = -2i + 7j.Find each spec...

Question

In Exercises 27–30, letv = i - 5j and w = -2i + 7j.Find each specified vector or scalar.||-2v||

Accepted Answer

Hey, everyone in this problem, we're asked to find the magnitude of negative seven A where A is the vector that's equal to I minus 11 J. We're giving four answer choices. Option A 122 multiplied by the square root of seven. Option B seven multiplied by the square root of 122. Option C the square root of in option D, the square root of 5987. So we get these double lines around our negative seven A indicate the magnitude of this vector. OK. So we're gonna get to that magnitude in just a minute. But let's start by calculating what that actual vector is. Let's calculate negative seven A. OK. So negative seven A, we have a scalar multiply multiple. Sorry, multiplying our vector A, we have a scalar multiple. All we're gonna do is multiply it just like we would anything else. So we have that factor of negative seven. We're gonna multiply it by our vector I minus 11 J and that scalar multiple is gonna apply to each component of a vector. OK. So we want to expand these brackets and apply this constant multiple to each term. When we do that, we're gonna have negative seven multiplied by I which gives us negative seven I and then we have negative seven multiplied by negative 11 J which is gonna give us 4 77 J. So now we have our vector negative 78, then that's the vector we're gonna find the magnitude of. So recall that the magnitude of a vector A in this case, the magnitude of negative seven A is going to be equal to the magnitude of negative seven I plus 77 J when we have a vector written in this IJ format. OK. Let's say it's written as X I plus YJ. Then the magnitude can be written as the square root of X squared plus Y squared. OK. So we're taking the square root of the sum of each component squared. Applying this to our problem. We get the square root of the X component or the I component, which is negative seven. So we have negative seven squared. Yeah, the square of the Y component or this J component which is 77 squared. If we simplify, this is gonna give us the square root to was which is the square root of 5978. Now, we want to simplify this. OK. This does not match with one of our answer choices. Just yet. So we want to simplify it if we can. And in order to simplify it, we wanna try to factor or break this value in our square root up into something that we can take the square root of nicely. OK. A perfect square. And then some other value and it turns out that we can write 5978 as multiplied by 122. Now, we know that the square root of 49 is equal to seven. So we can split this up further and write this as seven multiplied by the square root of 122. And now it's gonna be the magnitude of negative seven A that we were looking for comparing this to our answer choices. We can see that this corresponds with answer choice. B thanks everyone for watching. I hope this video helped see you in the next one.

In Exercises 27–30, letv = i - 5j and w = -2i + 7j.Find each specified vector or scalar.||-2v||

Key Concepts

Vector Magnitude

Scalar Multiplication of Vectors

Vector Notation and Components

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