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Remaining Iime: 594 57 (Min sec)Problamn 13.polnt) Give vector parametnc equation tor the Iine through the point (3,3) that is perpend cular t0 the line3t) :L(t)ple...

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Remaining Iime: 594 57 (Min sec)Problamn 13.polnt) Give vector parametnc equation tor the Iine through the point (3,3) that is perpend cular t0 the line3t) :L(t)pleue" answemsenueted Ansher pirvisw ResuitincorrectThis Enswer [ NOT conect

Remaining Iime: 594 57 (Min sec) Problamn 13. polnt) Give vector parametnc equation tor the Iine through the point (3,3) that is perpend cular t0 the line 3t) : L(t) pleue" answems enueted Ansher pirvisw Resuit incorrect This Enswer [ NOT conect



Answers

Find parametric equations for the line that passes through the points $P$ and $Q .$ $$ P(3,3,3), \quad Q(7,0,0) $$

In the question they're asking do Use the formula in exercise 12 4 45 Question to find the distance from the point to the given line. The point is 013. And the equation of the line is x equal to two t. by equal to 6 -2 t. And the equally tripolis city. Therefore in order to find the distance of the point from the line first we have to find yeah the equation of this line in a regular form. So this uh means that the normal victor for the line. Mhm is equal to the coefficients of the tv level. That is too -2 and one while the point that passes through the line yeah is equal to zero minus Blessed six and 3. So this is the point and the victors through the line. Now according to the question we have to find the yeah distance. So in order to find the distance we have to use a formula that is given in the exercise. The formula is the is equal to a vector crossed be vector divided by a vector. So the scalar of the cross product in the enumerate. And the scalar of the a victor. Where a victor is the normal to the line. Yeah. Mhm. Uh huh. And v victor is mhm. So be vector is the mhm mm victor which is contained in the line which can be found out by using the formula the coordinates of the yeah yeah of the line. Mhm subtracted bye. The point given him the question therefore Herbie vector is equal to yeah. Yeah. Mhm. Yeah zero minus zero cuomo one minus six comma three minus three. That is yeah that is equal to zero comma minus five comma zero. Therefore this is the v victor. So after getting the values of A. N. D we can find. All right. The okay. Cross product of A. N. B. According to the formula. This is equal to the table I. J. Key and the values of A is 2 -2. 1. That is the direction vector of the line given in the question and be as though victor. That is yes derived from the two points of the line. And the point given the question that is 0 -50. And therefore it is equal to cap five minus Jacob zero plus kick up minus 10. So this is equal to 50 -10. Now in order to find the distance from the point 013 to the line given in the question. It is equal to scalar of the cross product of A and B divided by the scalar of a victor. So this is equal to the numerator. The scalar value of the cross product will be five square plus zero square plus 10 square divided by this killer value of the director. That is route under two square plus two squared plus one square. That is equal to route under 1 25 divided by route under nine. Now this is equal to five. Route five divided by three. Therefore the answer to this question is the distance mhm. Of the .013 to the line given in the question has a distance V is equal to five. Route 5 divided by three, and this is the required answer of the given question. Mhm.

Here. I want to find the line equation in vector form that goes through these to coordinate points to do that. I call the first one, P second one Q. The first step is to find the line direction to do that. Just work out the vector PQ. It's a line joining, peter Q of course gives line direction and that will be minus 263 other words O que minus the vector 04 or five or P. So PQ then is Tokyo minus Opie, which is that Which is -2, two and minus two. No, every straight line you can write in the format R equals a plus tee times vector B. Where A is the point on the line and be his line direction. So for a I can use either P or Q. Doesn't matter or choose P. It's a bit simpler. So we have them are equals actor. OPM 045 plus T and B. Is the vector PQ Which is -2-2. Now I could make that simpler by just dividing bye to hear because direction could be any multiple of this. So I could write down here negative 11 negative one, awesome, correct. Here then I will have 0 -2 T. The I component four plus to T. Perhaps the J component & K -2. T. That's your k. Component and that's one possible answer. As I say you could have put time here minus one T. Four plus t and five minus t. Still correct

Here. I want to find the vector equation of a line that goes through the point 6 -5 2. And it's parallel to the Vector 1, 3 -3. No electoral equation has this form R equals a plus lambda B. Where A. Is a point on the line and be the line direction. So here Point is 6 -52. Look I can write if I want to as a column vector just bit neater. And Be here is one three negative three. That's the vector equation of the line which you can write Also as R equals six plus lambda, I plus negative five. Last three lambda jay plus two minus three lambda. Okay now that's in vector form if I want it in parametric form all I need do is do this. This could be X. Y. Z. Our X, Y. Z. Just read off values so we have X equals six plus lambda, Y equals negative five plus three lambda. Mhm. And the z component Will be 2 -3 λ. And that's the answer. In parametric form.

So we want to use the formula D equals a cross be magnitude over magnitude of A. Where is Q. R. And B is QP. So if we calculate Q are based on the points, we see that Q. R. Is the vector to negative 21 And QP is the vector 0 -50. Then we take their cross product. And in doing so we get the vector 50 -10. With that we want to take the magnitude, so we take the square root of the sum of squares and that gives us five Route five. So now we have this which is um a crossbeam magnitude. Now we just need the magnitude of A. Which is going to be three. So our final answer for D is going to be 55 over three and that's approximately 3.73.


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