Scalar multiplication: component form

what I want to do with this video is give ourselves some practice and hopefully some intuition on multiplying a scalar times a vector now what am I talking about when I say multiplying a scalar times a vector well let me set up a little two-dimensional vector here so let's say I have the vector W and let me give it an X component let's say it's X component is 1 and it's Y component is let's say it's 2 and I could draw it if I like actually that's a good idea it's always nice to be able to visualize these things let me get some coordinate get a coordinate axis here so that's my x axis that is my Y axis and so if I were to draw this vector in kind of standard form I would put its initial point at the origin and then its terminal point would be at the point 1 comma 2 so its x-coordinate is 1 its y-coordinate is going to be 2 so 1 2 this vector is going to look like is going to the initial point is right here it's terminal point is going to be right over there the vector in standard graphing it in standard form or visualize it in standard form would look like that now of course I could have the same vector and I could shift it around as long as I have the same length of the arrow and it's pointing in the same direction but if you know no one tells you otherwise it's nice to just put its initial point at the origin now let's multiply it by a scalar now what do we mean by a scalar well a vector is something that has a magnitude and a direction a scalar is just something that has a magnitude you could think of just the numbers that you started learning when you were 4 years old that those are those are scalars so for example we can think about what is what is 3 times W going to be 3 times W 3 is a scalar W is a vector now the convention we use for multiplying a scalar times a vector is you just multiply each of the components times that scalar so this is going to be equal to this is so R we have a 1 and a 2 and we're going to multiply each of those time's the 3 3 times 1 and then 3 times 2 and so this is going to be equal to this is going to be equal to 3 times 1 is 3 3 times 2 is 6 and so we see the resulting vector 3 we could call this vector 3w or it's going to have an X component of 3 and a Y component of 6 so if I were to draw it in standard form here X component 1 2 3 and then Y component 2 3 4 5 and 6 and so it's going to look like this let me see if I can draw it reasonably it's going to look like this and obviously I'm hand drawing it so it's not going to be exactly right but it's going to look like that so this is the vector 3 times W now notice what happened when I multiplied it by the scalar the direction didn't change but the magnitude did and you see what the magnitude changed by it changed it increased by a factor of 3 this was our the length of my blue arrow now is 3 times the length of my magenta arrow let's do another example I'll use the same vector W since we already have it set up let's multiply it times a negative scalar so let's say let's see what happens if I multiply negative 2 x times W and like always I encourage you to pause the video and think about what this would be and even if you have the time graph it out well we would multiply each of our components by negative 2 so it would be equal to negative 2 times 1 it would be the X component and the y component would be negative 2 times 2 and so this is going to be equal to the vector negative 2 comma negative 4 well let's plot that so the X component is negative 2 so it's negative 1 negative 2 and then the Y component negative 1 2 3 I'm going a little bit off of my axis 4 so that would be negative 4 there that's negative 2 so it's going to look something like the this it's going to look something like something like that so that right over there is a vector negative 2w now let's think about what just happened well we because we had the negative year it essentially flipped the direction by 180 degrees it's going in the opposite direction but one way to think about you could they kind of would still sit on the same line so the negative just flipped its direction if this if you if you consider whatever direction this was the the magenta vector W is going it's not going to go in the opposite direction and then it also scaled it up by two this is twice as long has twice the magnitude of our original vector and it's going in the opposite direction because of the negative sign so hopefully this gives you a little bit of intuition of what it means to scale a vector I mean literally the word scalar has the has the let me write it down ski scaler these are sets a scalar that's a scalar it has the word scale in it it's literally just scaling the vector it is changing its magnitude it might flip it around because of a negative sign but it's essentially changing its magnitude scaling it up or down or flipping it around with a negative sign