Scalar Multiplication of Vectors

Transcript

in this video we're going to talk about the scalar multiplication of vectors and what exactly is scalar multiplication when it comes to vectors here it is K times V is equal to Ka times the unit Vector I Plus KB times the unit Vector J now you might be wondering what is this V is a vector a vector is something that has magnitude and direction so for instance force is a vector you can apply a force of 100 Newtons east west north south so force is something that can have magnitude and direction so that's v v is a vector k is a scalar quantity a scalar quantity is something that has magnitude but no Direction for instance temperature is a scalar quantity it can have magnitude if it could be 70 degrees Fahrenheit it could be 90 degrees Fahrenheit but it doesn't have Direction you can't say it's 90 degrees Fahrenheit West that doesn't have that doesn't work so the scalar multiplication of vectors what it really is is just a number multiplied by a vector so for instance let's say we have the vector v and let's say it's 3i minus 4 J if we put a 2 in front of it we've multiplied the vector by the scale of quantity of two two is the scale of quantity so 2v would be 2 times 3i minus 4J which will give us 6i minus h a so you can multiply a vector by any number and what's going to happen is it's going to change the length of the vector so let's make a graphical representation of V so if we were to plot it think of i as the x value and J as the Y value so a vector that's 3i and negative 4J to draw when these travel three units to the right four units down and then the hypotenuse of this triangle would represent the vector v so this is V now if we multiply V by 2 we're gonna it's gonna double in left so that's 2v so that's the scale of multiplication of vectors now what about negative 3v what's the value for that negative 3v is just going to be negative 3 times vector v which is 3i minus 4J so it's going to be negative 9i plus 12 J here notice the X component is positive and the Y component is negative now because we multiplied V by a negative value X is now negative Y is positive so what's going to happen is the vector is going to change direction so it's going to be three times long but because of the negative sign it's going to go in the opposite direction so that is negative 3v so let's work on some practice problems let's say vector v is 2i minus 5j and Vector W is negative 3i plus 6 string go ahead and find the values for these vectors so 4V and negative 7w go ahead and multiply each of those vectors by the corresponding scalar quantity so for this one 4 values is going to be 4. times 2i minus 5j so it's going to be 8i minus 20 J as you can see this is not too difficult for the next one we have negative 7w so it's negative seven times negative 3i plus 6 J so we need to distribute negative seven times negative 3i that's positive 21. I and then negative seven times six strain is going to be negative 42 J so that's basically it for this video so now you understand what is meant by the expression scalar multiplication of vectors basically you're multiplying the vector by a number like 2 3 or negative 5.