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Point) The points (2,1,-5) , (1,3, 14), and (3, -5, -4) lie on a unique plane. Use linear algebra to find the equation of the plane and then determine where the lin...

Question

Point) The points (2,1,-5) , (1,3, 14), and (3, -5, -4) lie on a unique plane. Use linear algebra to find the equation of the plane and then determine where the line crosses the X-axisEquation of plane (use x, V, and z as the variables):Crosses the X-axis at the point:

point) The points (2,1,-5) , (1,3, 14), and (3, -5, -4) lie on a unique plane. Use linear algebra to find the equation of the plane and then determine where the line crosses the X-axis Equation of plane (use x, V, and z as the variables): Crosses the X-axis at the point:



Answers

Sketch the plane represented by the linear equation. Then list four points that lie in the plane. $$5 x+y+3 z=15$$

So in this problem we're given this equation and were asked to sketch the plane. Yeah, representing this equation. And then are asked to list four points four points on this plane. All right. So, first of all the way to sketch this is two. Figure out the X coordinate when Y and Z or zero, and then the Y coordinate went X and Z or zero. And the Z coordinate went X and Y are zero. In other words, the three intercepts that happened well, when Why? and 00 and I have two, X equals 12. And so X equals six. So that means I have 600. So that's one of the points on this plane in it. So, let's list that over here. That's one of the points on this plane. And that means on the x axis here and go 123456. So, I'll have point there. Okay, then, when X and 00 I have three, y equals 12. So why is four? So now I have another point here at zero for 0. Okay. So why is four so 1234 at that point there? So that means the plane's gonna intersect that line. I can do a little bit better drawing that line. Let me see here. Yeah. Almost there we go. And intersect that line. Okay? So now when X&Y are zero four z equals 12. one x 4. See his three. All right, so there's another .003. All right, so 123. No they're sick that point right there. So oh that missed pretty badly. Let me redo that. What's that one? That sign And then there's that sign. So there's the plane drawn there for us and now we need one more point. All right, So let's let x. equal one and y equal to what happens when X is one? That means I have two times one is two. Was Why is 2? So three times 2 is six Plus four. Z equals 12. Six plus two is eight plus four. Z equals 12. It's attract eight four. Z equals four, So z equals one. So here's the 4.1, two, one. There's four points on our plane and our plane is drawn over there on the left hand side.

According to the ocean, We need to find decoration off complain. We have given the plane bossing through the point. The points are 531 and one to the minus fight and minus one to the fun. We know that equation off the plane passing through three known Killing Your points equation The plane bossing true three. Known calling your Blaine's He's x one y 10 even ex toe by toe. Is that toe X three by 303 Then we have three known Kalin. Your points we know we use dominate do dominates matter to find out though Value off equation off plane when the u determinate matter x minus Excellent value minus y even Jed Mine, Sir Devon X two minus excellent by two minus y one that to minus Erdogan Extreme minus X one by three Minus y one jag three minus red even is equal to zero When we substitute this value in the equation, we have X minus fight by minus 30. Zed minus one and one minus. Fight three minus three minus five minus one minus one minus five three minus zero one minus one. First we simplified these. They dominate you know, simple form X minus Fife. Value minus three. Zegra minus one minus 40 minus six minus 60 zero. They're easy toe Find order. Determinate. When video dominate, we eliminate with the first row X minus five Zito minus by minus three zero minus 36 Less zed minus one zero is equal to zero. When we simplify this equation, we get the value ease. 36 way minus 108 is equal to zero. And when we sold this equation we get the final result is why we could do today There is a question of dark plane.

We have given a portion three X plus two y plus toe plus through there is equal to 12 and we need to sketch the plane the president did by the lean Eric question that list full points. That's like end up it. First we need to skin the plan playing. Then when it list up, four point of is is flying this plane, so finding the points first. So for that, just taking some values at why is equal to zero Leo and is that is equal to zero so we can die. The X is equal to Otwell or from here we can get over Lou effects that is full. And the point will become the point. From here begin simply right, fool. And now that X is equal, does it'll? And that is equal, though. Zero. So we will get here Where Lou off. Why? Which is sick? The point will become zero six little Now what will we will do here? So at exit weirdo zero And why is it Porto zero? So we'll get the value of said so we can get here. That is a weirdo, say and point here will get 00 six. So now from this weekend, let's get a graph which is here. So just plotting this Lord, please. Three points on the three D coordinate plain. So we get up graph, We'll have a graph like this. So this is a graph. This is a light. So here, recondite possible point that can lead possible plan would be to 30 1st another 1203 zero for two. There are 24 This is a possible point we can see here.

Mhm. For the given problem, we see that We're gonna have the vector 5. 3 Euro. And then we have the point for -1, based on the parametric equations? Or is the symmetric equations. And then we also have 2 -3. I get five is the other normal vector. But we see that the intersection point between the plane and the line. If we plugged in this point, for example, We would end up getting four times five is 20 um negative one times three is negative three, so 20 minus three is 17. So this point right here does satisfy the given equation. Um and we see that since we have four negative one, it doesn't matter what the Z coordinate is. It could be an easy coordinate, but as long as we have four negative one, it will be satisfied.


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