5

8 IXWI 1 li { J W V 8 = 1 W 3 3 { 8 3 1 0: 4 W V H { 3 1 i' $ 3 V 8 1 3 8 D1D 1 1 3 J 1 3 8 2 8 1 { 8 W 3 3 & 0 8 { 3 ? 1 8 J1 8 { 3 0 g { 3 1 3 [ 8 j} 1 8...

Question

8 IXWI 1 li { J W V 8 = 1 W 3 3 { 8 3 1 0: 4 W V H { 3 1 i' $ 3 V 8 1 3 8 D1D 1 1 3 J 1 3 8 2 8 1 { 8 W 3 3 & 0 8 { 3 ? 1 8 J1 8 { 3 0 g { 3 1 3 [ 8 j} 1 8 2 [ 1 E 8 1 1 1

8 IXWI 1 li { J W V 8 = 1 W 3 3 { 8 3 1 0: 4 W V H { 3 1 i' $ 3 V 8 1 3 8 D1D 1 1 3 J 1 3 8 2 8 1 { 8 W 3 3 & 0 8 { 3 ? 1 8 J1 8 { 3 0 g { 3 1 3 [ 8 j} 1 8 2 [ 1 E 8 1 1 1



Answers

$$\frac{3^{2} \cdot 4^{5} \cdot 5^{3}}{5^{3} \cdot 3^{3} \cdot 4^{6}}=$$ (F). \frac{1}{60} (G). \frac{1}{12} (H).\frac{3}{4} (J)$$12$$

Okay, So today I'm gonna be showing you guys how to use the dot product to solve problems like this. So the problem, um, is usually given vectors to find three times w and V. Times Week. So the first thing we wanted you to want to do is to find the doctor permit. So the Doc products says that if you have two vectors, will call it a and B Ah, and you're multiplying them by each other, just like with Vivian W you're gonna have equal to a one multiplied by be one plus a two multiplied bye Be too. And this can continue on and on. So plus a three beat Lee plus a four before it all depends on the number variables you have in the equation in the, um, the definitions of the vectors. So in this case, we have only two variables. So if you look at V and W, we have I and J for both of them. So we're gonna have one, um, a definition of a dot product similar to this one. So you might be wondering what then is a one and B one and in the face more defining in terms of e and W. What's V one would still be one? So the first thing we want to dio is let's write this stop product in terms of Ian W So we're gonna have v times w equal Thio V one w on plus V two w two. Okay, So, back to that question. What is V one? What is W one? What's V to what's w too? So let's look at V. So the one is gonna be the first number in front of the first variable. So we have negative eight. I that is gonna be RV one. So negative eight. That's gonna be our V one. And when we're looking at what V two is, it's gonna be the next value. So negative three. And it's really important not to forget that negative sign. So that's gonna be V two them. We're defining it with W. We're gonna do the same. Thanks. And negative 10 is gonna be w one, and negative five is gonna be w two. So now we have those values that we need and we can plug him in so same thing the times w gonna be equal to V one. which we said was negative eight multiplied by W one, which is negative term plus I ve to which is negative three times W two, which is negative five. So if we multiply these out, we're going to get the Times w to equal negative eight times negative term and that negative signs gonna cancel out. So we're gonna have a positive value, and it's gonna be eight times 10 which is 80 plus. So we're gonna do the same thing, and we have two negatives again. So these the negative is gonna cancel out, and you're gonna be left with a positive value. So three times 5 15 And when you add those together, you're gonna get 95. Okay, Now, the second part of the problem says, What's the toms V? So we're going to do the same thing. We're gonna set up a definition of the dot product in terms of v times Be so it's a lot simpler than you think. That's not gonna be some crazy change. You're gonna do the same thing our first, our first vectors. We so v one and our second vector is vey as well. So it's gonna be V one times V one plus me too. Times be too. And we're gonna then put those values in Soviet times V is equal Thio Negative eight times negative. Eight plus the two times we too, which is gonna be negative three times negative. Three So and then all we have to do now is multiply them and add them together. So negative eight times negative. Eight of the night gives you they're gonna cancel out and you're gonna be left with eight times eight, which is 64 and same thing for the next part. Negative. Three times negative. Three negatives were gonna cancel out three times three is nine. So 64 plus nine is gonna get you 73.

This question is asking us to get a and B off solution. Awesome. These these air the equation from the problem. If he said set of metric from that, you can get thinks Hey, 10 Negative. Two five, two. This is equal to A and B. And if you served for this metric, you're gonna get one. You're 01 point fight and 0.25 At this point, I believe you can solve this metric for since this is all this problem is from s A T or a city. Oh, let me tell you almost vester way to solve this problem. So we're given to equation here. So Oh, at two equations making a hearing reader one variable with two means we can multiply Disa quiz. Um, Times five. This It's the color. So first equation Times five, then Sikander quiza times Negative three. So they will give us 30. Hey, close foodie. Be right. Five times eight. Give us 40 in the 4 to 5 times 5 25 and a times negative. It's gonna be negative. 30 here. Negative three times 10. A snake a 38 and here is will be positive. Very thick. You quit to be, is he? And these were called to negative 6 2/10 negative three. And you're adding this tree cuisine. So here is going to zero. Plus, you're always going to 76 in is equal to 19 and B it's gonna be 19 over 76. And if you almost if your simple simplifying more again one over four And so the answer is calling it, say, cause they only has 1/4 here. I mean, the answer, uh, multiple choice has only one before for a day. So the interest is a if, you know, make sure Oh, you can just look b you can plug be two one forest equation for a second equation. They will give us a You go to one head. So our answer this party is it

Eight Over to W-plus one Equals three over W. So I'm gonna do this problem a little bit different just to show you a different way of doing these type of problems. So this is called cross multiply when you have an equal sign and you have two fractions. So I take this times this and this times this and set them equal. So three times two, w plus one Equals eight times W distribute my three three times to W is six. W plus three equals eight. W Subtract six W. Because you want your Variables on one side. So 8 -6 is to W equals three And then divide by two. So you get W Equals 3/2 as an improper fraction Or one and 1/2 as a mixed number.

The problem we're told the matrix A. Is equal The 3 -22 zero on negative four. And negative three to negative form. And the Matrix B is a matrix 40 two negative one negative 13 Now first we want to find A plus B. Now in order to add two matrices they must have the exact same dimensions. Well a is a three x 3 makers But B is a three x 2. And so that means that this is not possible because they don't have the same dimensions. Yeah. We also want to find A minus B. Yeah this is something else that we can't do because these don't have the same dimensions. And so this is still not possible. Yeah. Now we can find to A to A. Just means take everything in A. And multiply it by two. So that top probably six. And then get a 44 Than 0 to -8. The -6 4 -2. Yeah. And then lastly we want negative three B. And so it just means to take everything and be And multiply it by -3. That's 1912 0. Then you have 63 3 -9. So that's our matrix -3 B.


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