Differentiating Unit Vectors and Scalars

University
Massachusetts Institute of Technology
Course
Multivariable Calculus
Pages

1

Academic year

2022
Author

Sporkz
Views

75

p {margin: 0; padding: 0;} .ft00{font-size:20px;font-family:Arial;color:#000000;} .ft01{font-size:18px;font-family:Arial;color:#000000;} .ft02{font-size:18px;font-family:ArialMT;color:#000000;} .ft03{font-size:18px;line-height:22px;font-family:Arial;color:#000000;} .ft04{font-size:18px;line-height:22px;font-family:ArialMT;color:#000000;} Unit vectors ● A unit vector is a vector with a magnitude of 1. A unit vector has the same direction as another vector, but its length is different. A unit vector can be scaled to anynon-zero dimension without changing its direction. ● The term unit vector is used to describe a quantity that has magnitude of one and direction. It is not the same as a scalar (a number without any direction). Here’s another question about vectors with the same direction that can be misunderstood.It is the problem of finding a vector with a given length in a given direction. Suppose we have to find a unit vector in the same direction as the vector with components[2,1]. However let’s review, what is a unit vector? v is a unit vector if its length is 1. How might we find a unit vector in the same direction as [2,1]? When I see vectors in the same direction, it tells me that v should be some lambda times 2comma 1. Now we can choose a vector in the same direction for each of these different lambdas. Letus choose lambda such that the length of v is 1. What will be the length? The length of vector v, which is lambda times bigger than thevector 2 comma 1. Thus, the formula for the length of [2,1] is lambda times the square root of 5. Now that we are familiar with the concept of lambda, we can find its value by taking 1 overthe square root of 5. Thus, the vector v is 1 divided by the square root of 5 times [2,1].