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Lety =Find the dislance from to the plane in R? spanned by u1 and u2The distance Is (Type an exact answer; using radicals as needed )...

Question

Lety =Find the dislance from to the plane in R? spanned by u1 and u2The distance Is (Type an exact answer; using radicals as needed )

Lety = Find the dislance from to the plane in R? spanned by u1 and u2 The distance Is (Type an exact answer; using radicals as needed )



Answers

Find the distance between the point and the plane (see figure). The distance $D$ between a point $\left(x_{0}, y_{0}, z_{0}\right)$ and the plane $a x+b y+c z+d=0$ is $$ D=\frac{\left|a x_{0}+b y_{0}+c z_{0}+d\right|}{\sqrt{a^{2}+b^{2}+c^{2}}} $$ $$ (-2,1,0), 2 x+5 y-z=20 $$

Welcome back to another cross product problem. This time we're looking at what happens if we want to find the distance from a point. Let's call it P to a plane. We know we can define a plane by three different points. Let's call these Q, R and S. And so if we define a couple of vectors, let's say a between Q and r. Be between Q and S. And say see between Q and r. Point. Then what we can do is try and figure out the distance from the point to the plane. Using some things that we know about. Triple products. Specifically the volume of the triple product of the parallel pipe ed between a B and C is defined as the magnitude of across B times the magnitude of C. Cosign theta. And so rearranging this a little bit. If we write C dot a cross B divided by the magnitude of a cross B, then this is going to be the magnitude of C. It's the length of that vector C times Cosine Theta where theta is the angle between our vector and perpendicular. This is theater right here. Now, since cosign Theta is adjacent over hypotenuse, that means the length of this perpendicular lines. The distance from the point of the plane, going to be the magnitude to see times. Cosine Theta. We can use this if we want to calculate the distance from say point P 214 to the plane divided by Q, R and S. In order to do that first, we need to calculate projectors defining the plane. So A is Q. R. AR -Q Sierra -1 To -00 0 and Q. S. That's s minus cues. Again zero minus one, zero minus zero, three minus zero. We'll also need to see that's the vector QP p minus q is two minus one. One minus zero, four minus zero. So if we want to calculate the magnitude of c dot a cross B, we can do that in one step using our triple product. If we plug in C. A and B into our matrix here, I'll point out this isn't the only way we can do this. Um But we'll talk about another way in just a second. So let's plug in 114 -1-0 And -103. And using the formula from our textbook, remember we ignore the first column And look at two times 3 minus zero times zero, Going to be 6 0. And normally we would multiply by I We're actually gonna multiply by one minus and ignore the second column -1 times three zero times negative one. Eight of three minus zero times not J but times one again us. And then we ignore the third column negative one times zero minus two times negative one will be zero minus negative too, Which is 0-plus two times four. And since we don't have any eyes jay's or k's this is not a factor but just a number six times one minus negative three, That's plus three times 1 Plus two times 4. And so we're looking at six plus three plus eight is 17. The other thing we need from our formula remember is the magnitude of a cross B. And so we can get that by calculating the cross product of A and B. Once again we're looking at six minus zero, I minus negative three minus zero jay plus zero minus negative too. Okay, Giving us the vector six three two. If you want to calculate the magnitude of a Crosby, that'll just be the magnitude of 63 two. Which is the square root of six squared Plus three squared plus two squared or The square root of 36 plus nine plus four. And that's the square root of 49 or just seven. Since we determined that the distance is the ratio of the two numbers that we just found, we get the distance is let's go back to that cross product Or the triple product with 17 over seven. I said earlier, this wasn't the only way we could have found this. Since we already know the cross product a Crosby, you could have the magnitude of that and then found the magnitude of C. Got a cross B. Just using the vector C. And a dot product. That would avoid having to do the cross product twice. Thanks for watching.

Cushion here where we could not given the pond X zero y 00 and, uh, equation a plan i x plus b y plus he set plus D E Co +20 that we can find distance between them equal to the absolutely x zero plus b. Why zero policies that zero plus city divided by the square root on the I X square, a square plus B squared plus C square in this question here were given upon will be the origin on the plan will be eight. X man is far. Why, plus the co +28 notice. Then we can bring the it to the left hand side. And then we put the coaches around here and Dan from here we can find a distance. D e co two absolute value here eight times zero minus four times zero plus zero minus eight absolute value divided by the square root under a square plus minus four square plus one square. And then we can simplify together the top. We go to the eight for the bottom, where the square root off the 64 plus 16 plus one, you go to the 81 hour again. The answer. It go to the 8/9

In the question here would record that, given the point x 000 and the equation the plan be explosives. Is that plus D E coaches zero that we can find a distance between them? It will echoed you. The absolute value I x zero plus b y zero plus is that zero plus D dividing bind his squared plus B squared plus C square in this question here were given the point. You go to the 321 and a plan here will be X minus y plus to set equal to far notice that we can bring the fall to the left hand side and equal to the zero here and then by formula, we can get a distance between them equal to the absolute under three months two plus two times one minus four divided by the square root under one square, plus minus one square, plus to square. It will certainly find the top. We get equal to that three months for equal to minus one. Getting what you want. Now the of anti matter squared off the wondrous one to bless far to six. Then that's gonna be the answer. One of us got it down to six

In this question. Very conduct given the Pont X zero y 00 and the Plan I X plus B y plus sees that plus D equals zero. Then we can find a distance day between them. Could you after him? That extra plus B y zero plus is that their a plus day, the value by the square root of a squared plus B squared plus C square in discussion with When the pond ICO tuna three to minus one. And the plan will be the two x minus three y plus for that equal to 24 by formula, we can find a distance between them. You go to the absolutely 3 to 10 3 minus three times to and then blessed four times, minus one minus 24 here dividing by because soon you can understand to the manor standing for equal to zero. And then we divide by square root on the two square plus Ministry square plus four square. Now we see the top. We say we can consider this with this one on minus four minus 24 we equal to the miners 28 equal to 28 divided by the square. It off the Here we have the four plus nine plus 16. The coaches square with 29. So that's gonna be the answer, Yeah.


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