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Addition and Scalar Multiplication of Vectors - Concept

Teacher/Instructor Norm Prokup
Norm Prokup

Cornell University
PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

The addition and scalar multiplication of vectors are basic operations that can be done using their geometric or the algebraic representations. When looking at the geometric representation we can understand scalar multiplication of vectors as scaling. The addition of vectors can also be performed two ways using the geometric representation.

Now that we have the concept of components of vectors we can redefine the way we add vectors and we can also introduce another operation called scalar multiplication.
Adding vectors algebraically, let's say we have two vectors given to us in component form so u is u1 comma u2 and vector v is v1 comma v2. What would be their sum be? Well the sum is u1+v1 you add the first components and then u2+v2 you add the second components so you're adding the two vectors component wise.
Now I also want to introduce the idea of the zero vector. The zero vector is a vector with components 0, 0 has 0 length. This vector has the property you can add it to any other vector and get that vector back so u plus the zero vector is u and the zero vector plus u is u.
Vectors act in many ways like real numbers in terms of their Algebra but they're not identical to real numbers in terms of their Algebra but they're a little bit different. One of the differences is, multiplication of vectors is a little harder and so the first kind of multiplication I want to talk about is scalar multiplication. Now you remember that a scalar is a quantity that only has magnitude not direction so we're going to be multiplying vectors by scalars and let's go look like this, if k is a real number and u is some vector u1,u2 then the scalar multiple k times u is going to be defined as ku1, ku2 so you just multiply the scalar through each of the components kind like distribution.
Let's see what scalar multiplication does in an example so let's say u is -3, 1 and I had actually graphed this vector here so -3, 1 looks like that. What would 3 times u be? Well according to this definition I multiply the 3 inside and I get -9, 3 times 1 3, so that's 3 times u. -2 times u, I multiply -2 times -3 and I get 6, -2 times 1 -2, 6, -2 and 0 the scalar 0 time s vector u is going to be 0 times -3 0 and 0 times 1, 0, and that of course is the 0 vector, so the scalar zero times any vector gives you the zero vector. Just keep in mind that these two zeros are different this is the the real number 0 and that's the vector 0.
Now just to let you know what these some of these vectors look like. Let me plot -2 times u. Its components are 6, -2 so I go 6 to the right and 2 down so it's going to end here, so notice that this vector is going to end up being twice as long as u so this is -2 times u it's twice as long but it's in the opposite direction it's opposite because the coefficient k is negative in this case and that'll always happen whenever you multiply a vector by a negative number you'll reverse its direction so scalar multiplication can lengthen or shorten a vector and it can reverse its direction but if we multiply by a positive constant we always get a vector in the same direction.

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