![]() ![]() So you can write it asĥ square roots of 3 i plus its y component. Written as a multiple of the x unit vector. What's its x component? Its x component can be The sum of its x component and its y component. So how can we write vector v? Well, we know the vector v is And what multiple is it? Well, it has length 5, Multiple of this vector j, of the unit vector j. So the y component goesĬompletely in the y direction. So we could say v sub y, the yĬomponent- Well, what is sine of 30 degrees? Sine of 30 degrees is 1/2. Similary, we can write the yĬomponent of this vector as some multiple of j. And that's equal to- cosine ofģ0 degrees is square root of 3/2 -so that's 5 square Does that make sense? Well, the unit vector i goes in You don't confused -times the unit vector i. ![]() ![]() times the unit vector i- let me stay in that color, so To 10 cosine of 30 degrees times- that's the degrees Switching colors to keep things interesting. Square roots of 3, or something like that. This has a magnitude of 10Ĭosine of 30 degrees. Of that unit vector? Well, the unit vector hasĪ magnitude of 1. So v sub x is in the exact sameĭirection, and it's just a scaled version of It has a magnitude of 10Ĭosine 30 degrees. Multiple of i, of this unit vector? Well, sure. Vector, and then you put this tail to this head. When you add vectors, youĮssentially just put them head to tails. ![]() So hopefully knowing what weĪlready know, we knew that the vector, v, is equal to its x component plus its y component. When you add vectors, youĬan put them head to tail like this. The sum of its x component plus its y component. So how does that work? Well, this vector here, But if we're dealing in twoĭimensions, we can define any vector in terms of some sum And as later we'll see in threeĭimensions, so there will actually be a thirdĭimension and we'll call that k, but don't worry about So why did I do this? Well, if I'm dealing with And that is the same thingīut in the y direction. It has magnitude 1 and it'sĬompletely in the x direction. The imaginary number sense, you should realize that that's See this i without the cap, and it's just boldface. We denote the unit vectorīy putting this little cap on top of it. It just goes straight in the xĭirection, has no y component, and it has the magnitude of 1. Writing an Arabic or something, going backwards. Is no different than what we've been doing in our physics Mind, this might seem a little confusing at first, but this So what does that mean? So we define these Something I call, and I think everyone calls it, unit Having to always draw a picture of representing Need a coherent way, an analytical way, instead of Linear algebra, where we do end dimensional factors -we Of vectors, three-dimensional vectors, or we start doing Vectors- and maybe we're dealing with multi-dimensional Problems- But once we start dealing with more complicated But what I want you to do now,īecause this is useful for simple projectile motion But we've reviewed all of that,Īnd you should review the initial vector videos. Through SOH-CAH-TOA and say, well, the sine of 30 degrees Think is square root of 3/2, but we're not worried about X would be 10 times cosine of this angle. Second nature of how we would figure these things out. The y component of the vector, and it would have Would use a notation, v sub x, and the v sub x would haveīeen this vector right here. I always gave you a vector, like I would drawĪ vector like this. We launch something into- In the projectile motion problems, Both define the same vector, the vector notation is simply cleaner and more easily worked with. this is simpler to work with than the magnitude degree calculation of magnitude is sqrt(5sqrt3^2+10^) and direction is arctan 10/5sqrt3. Since you already need to break the vectors into components to solve the problem, 5sqrt3 ihat+5 jhat, and 0 ihat+5jhat, you can present the vector as 5sqrt3 ihat+10 jhat. Add a vector with a magnitude 10 at 30 degrees to a vector of magnitude 5 at 90 degrees. It is a shorthand way of writing out the individual components of a vector, which becomes very useful when manipulating multiple vectors. The plus and minus are not operators when you use a comma and parentheses like this, and that is where you need to bring your brain. To visualize what I'm saying, try using a comma in the unit vector notation: (+5) i hat, (-3) j hat. You need to forget about the plus or minus sign as an operator in this and think of it as an indicator of positive or negative values. The i-hat, j-hat, k-hat etc don't just disappear and allow you to add the values to get the magnitude of v. I think what you're missing (it isn't well explained) is that unit vector notation is an alternate way to define a vector, not an actual equation. ![]()
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