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For all problems on this page, use Ui = (~1,~2) andv = (1,-3). points) Find compi v(5 points) Find proj , u (Note the different order than part (a)I)(5 points) Eval...

Question

For all problems on this page, use Ui = (~1,~2) andv = (1,-3). points) Find compi v(5 points) Find proj , u (Note the different order than part (a)I)(5 points) Evaluate the expression |lzu 3v1l(5 polnts) Find two unlt vectors that are parallel to 0,

For all problems on this page, use Ui = (~1,~2) andv = (1,-3). points) Find compi v (5 points) Find proj , u (Note the different order than part (a)I) (5 points) Evaluate the expression |lzu 3v1l (5 polnts) Find two unlt vectors that are parallel to 0,



Answers

Find vector and parametric equations of the plane that contains the given point and is parallel to the two vectors. Point: (0,5,-4)$;$ vectors: $\mathbf{v}_{1}=(0,0,-5)$ and $\mathbf{v}_{2}=(1,-3,-2)$

So in this question, were given two points. The first is Point P, which is at five. Well, let in 12. And then the second is the point. Q, which is at 1 14 13. And so now we're interested in finding the vector Peeps you So the vector p Q is just gonna be that the change in the X coordinate frumpy Dickie The change in the White Corner from Pedic You and changing the Sikh ordinate from Pete Acute Rights. We look, um, from P to Q. The X coordinate changes by negative four, right from P to Q. The y coordinate changes by positive three and from P to Q of the Z. Coordinate changes by one increases by one. Um, keep saying from P to Q because that is, uh, rather important. Um, if we're going the opposite way from cute API, we would have the same, um, type of defector. It would have the same numbers, but of being the opposite direction to be positive for negative three negative one right in the direction PDQ acute API matters right, because vectors are quantities with both direction and magnitude. So you know the magnitude doesn't change when we switch, Pinky, the direction does so right now what we're interested in in just the vector P Q. Um, and this is when we express it. We might also express it in terms of the unit vectors I, j and K. So we could also call this peak you negative. Four times the unit vector I plus three the unit Vector J plus Unit Vector K. This is just another way of expressing it. I is the unit vector 100 It's so we multiply something when you buy. I buy some scaler. A multiplier like negative four were really just talking about negative four in the X direction. Same JJ is the unit vector 010 So multiplying J by three is really saying just three in the Y direction and in probably guess that K is 001 So much wine k by. Anything would just be that thing in the in the Z direction. Right? So these two are equivalent ways of expressing how the vector p Q So cool. Now we can move on, Teoh finding the magnitude of the spectre on. We do know that this is it. This is an exercise that we've done before. The magnitude of the vector P Q. Is going to be the square root of the son of the squares. Other components, right, So we have negative four squared plus three squared plus one square. Um, all right, this is the X Y and Z components all squared some together, and we take the square root of that to find a magnitude. This is immediate from the by figuring in theory. And we see you know, this a squared plus B squared equals C squared were just generalized into the third dimension. Um, so and from this, you can kind of tell that we're finding the distance between P and Q by finding the high pot news of, ah, a right triangle of sorts. So when you do this, we get 16 plus 9 +123 of the square root of 26 and now 26 is two times 13 so we can't reduce this anymore. It's just gonna be the square root of 26. But this is the magnitude. So he found the magnitude of 26 now we can use this magnitude to create a unit vector So I'm gonna copy down my vector in a vector. Waas Ah, negative for 31 And if I wanted to make a unit vector, I'll call it you. And for one week, a unit vector in the direction of Peak You What I do is a teak he que and divide it by its magnitude. This way it ensures that the magnitude of of P Q is equal to one of the unit Vector in the direction of P Q is equal to one. So if I take a vector negative for 31 and divided by the square to 26 I could express it like this, or I could actually do the scale multiplication. And we get scored a four over very negative for the square 26 three over the square, 26 and one over the square to 26. So this is, um, this is a unit vector in the direction of peak you. But it's not the only unit vector in the direction of Peak you. There's also, um, what we might call me anti parallel direction. So, uh, if this is P Q unit vector that we just drew was this right This has a magnitude of one, whereas peak you have magnitude of square to 26. So we found the manatee. The doctor would magnitude one in the exact direction of peak you. But we could also find the vector magnitude one in the opposite direction in the exact opposite direction. Like this this red evacuated I drew. We could still make it of magnitude one, so it would still be a unit vector and would still be parallel Torrey uh, vector Peak. You just be in the opposite direction and we call this anti parallel 90 parallel. So it is parallel because if he's too, ventures were existing in space and he translated them. They would never intersect, but they are moving in opposite directions. So we call this the anti parallel. And, uh, we might write that as you Q p right. Instead of going from P to Q. Now, we're kind of going the direction of q p. Um, and let me talk about before about here with the direction from P to Q or Q two p. Um, what we've done is just gone in the negative direction in all of our components, right? So Q P is just the opposite of peak you. In that same way, the unit vector in the direction of Q P is the opposite of the unit vector in the direction of P Q. So we'll have to get through this whole process again. All you do is take negative unit vector in the direction p. Q. And that would just be positive. Four over the square, 26 negative three of this word of 26 and negative one over the square to 26. So both of these vectors are unit vectors, and they're both parallel to peak you. But they are in. They are anti parallel to one another. They are moving in the opposite direction. So these are both of the directors that are parallel to P. Q. And we found them by finding the magnitude and then making a unit vector out of the Becker Peak. You by dividing, uM, p que by its home magnitude

In this question. Where did you find the question? Underline Sustained in going through the point to jump on far three Pawn five. And in this parallel, June the Defector three I plus J Man, stay in it for this factor. We can register into the victor from with a three to minus scorn. And here the 1st 1 want to fight the fat, The a question form underline. And then it would be Ah in we coach you the point him will be you jump on far trip on five then plus the vector No Russian have a B three to minus one. Been tempts the very body here and now in terms of the brand Which question then we should have The X now will be the first coordinate. Now that we have job last three day for the why we attend the second coordinate with the tube on far plus Judy and fun is a pretender. Last coordinate. It will be the trip on five minus de

Okay, so we want to half find this factor PR. So we'll take the two minus negative four and the six minus one that will give us the pure Vector 65 And we want to find t uniph actors parallel to this. So we need to find this unit factor so we'll do our vector 65 over the magnitude. What's your magnitude on the side? Six square plus five squared. So it's a square to 61 and we'll make our uniph actor, then six. Over. I would write that as one over the square to 61 times the vector 65 We can do that and we'll have the same direction positive and a negative for another one for opposite direction. And these would be parallel because there are along the same, uh, same trajectory as the factor 65 just with a different length.


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