In Exercises 39–46, find the unit vector that has the same dir...

Question

In Exercises 39–46, find the unit vector that has the same direction as the vector v.

v = 3i - 2j

Accepted Answer

Welcome back. I am so glad you're here. We're told, given the following vector V which is equal to 11 I minus seven J obtain the unit vector that aligns with the direction of V. Our answer choices are answer choice A U equals 110 I minus 70 J. Answer choice B U equals square 70 divided by 11 I minus square at 1 70 divided by seven J. Answer choice C U equals 11 divided by 70 I minus seven divided by 70 J. And answer choice D U equals 11 square up 1 70 divided by 1 I minus seven square up 1 divided by 1 70 J. All right. So when we are finding the unit vector, we recall from previous lessons that that is taking our vector and dividing it by its magnitude and we have our vector that's given to us. So in the numerator, we'll have 11 I minus seven J. But for our denominator, we need to know the magnitude of our vector. And we recall from previous lessons that for the magnitude of our vector, when our vector is given in terms of I and J, it's going to be equal to the square root of A squared plus B squared where A is the coefficient of I and B is the coefficient of J. So we have the square root of our A that needs to be squared A is the coefficient of I. So that's 11, we have 11 squared plus. Now, for B B is the coefficient of JB is negative seven. So negative seven squared and now it's just simplifying 11 squared is and plus negative seven squared is positive 121 plus 49 is 170. So this is the square of 170. And that's what goes in our denominator. Now, it's just a matter of simplifying. We don't want a radical in our denominator. So we're going to rationalize this multiplying both the numerator and the denominator by the square root of 170. Now, we can distribute the square root of 170 in the numerator. So that will be 11 square rut, 170 I minus seven square rut 170 J all divided by, in the denominator, the square root of 170 multiplied by the square rut of 170 is 170. And that's fine. But we can also separate this into two separate fractions which would be 11 square rut 1 70 divided by 1 70 I minus seven square root, 1 70 divided by 1 J and that's the one that matches the way our answer choices are formatted. We look at our answer choices and this matches with answer choice. D well done. We'll catch you on the next one.

In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 3i - 2j

Key Concepts

Vector Components and Notation

Magnitude of a Vector

Unit Vector and Normalization

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