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(a) Find a nonzero vector orthogonal to the plane through the points $ P, Q, $ and $ R $, and (b) find the area of triangle $ PQR $.

$ P (1, 0, 1) $ , $ Q (-2, 1, 3) $ , $ R (4, 2, 5) $

A. $<0,18,-9>$ or $<0,-18,9>$

B.$A r e a=\frac{9 \sqrt{5}}{2}$

Vectors

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Hello. Welcome to this person. And this lesson we'll find an underwater. That is the passes through the point p ko out and it's going out to the plane. Good. So that is the victim, p k l across the victor p R. Right. So the first one is this. We'll need the PQ that is sick or two. So negative two mhm negative. Two minus one F one minus zero. Then we have 30 minus one. Okay, so that as the victor, they get to three. I last j then plus two K. We equally need Director P. R. So that is four minus one. Yeah, two minus arrow than five minus one. So that is the victor today. I lost two j. Yeah, last four K. Yeah. So means that the p kill cross p. L would become the determinant maitresse with columns for the increase of Victor. Peaky O negative three woman two. And for that of the p r, we have three, two and four. Okay, Mhm. Uh huh, Yeah. Mhm. So let's expand that Using the cough, the bottom metal. So the I've but we delete this thing of 12 24 Yeah, minus G So the minus j wittily the first rule. The second column so that we have negative story 32 and four. For the cave back, we delete the first rule in the last Colombia third column. So we have negative 33 one and two. So this becomes four times one minus two times two the minus Jerry of negative. Three times four minus, three times two. Then they give bad. Yeah, negative. 38 times two minus three times one. So that is three. So this is zero I minus so 12. Negative tool. Yeah. Mhm minus six. That is negative. 18. So negative. 18 plus 18 minutes. Negative. 18 times negative. That is positive. 18. And here you have negative six minus three. That is negative. Nine key. Okay. Yeah. So we have the victor. That is also go now to the plane. As p, uh, course kill. That is the call to 18 J minus nine key or the negative of it. Yeah, So it can call a B PQ Cross p. Oh, that is he called to I get to 18 g plus. Yeah, last and nine k. Okay, then the next question is to the keypad is to look for the triangle. The area of a triangle P Oh, two mhm more. Okay, so area mhm of the strangle is recalled to half of the magnitude of this vector that we just found. Mm. Okay, so we have squared off 18 squared glass, nine squared. Okay, All over to mhm. So this gives. Yeah. So 18 square gives three, 24 then nine squared. Gives 81 so that the spirit of mhm Yeah. Yeah, very. Uh, 34 to us each. 14 that is 405 What? Uh huh. Okay. 4 35 or their five divided by one. Okay, so actually turns out that you can have 81 times five in this 1722 So that you have screwed of 81. Spirit of five. And there is nine. Spirit of five on two. So this is the area. The area? Yeah. Paul, for the triangle. P. Yeah. Kill. Wow. Okay, Potential time. This is the end of the lesson.