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For the pair of vectors A (8.00i (a) The scalar productO0j) and (9,00i0Oj) in the xy plane_ determine the following (Enter all angle answers between and 1809.)(b) ...

Question

For the pair of vectors A (8.00i (a) The scalar productO0j) and (9,00i0Oj) in the xy plane_ determine the following (Enter all angle answers between and 1809.)(b) The angle between the vectorsc) The angles and 8 which are respectively the (smallest) angles between the vector and the positive and positive axes(d) The angles and $ which are respectively the (smallest) angles between the vector B and the positive and positive axes

For the pair of vectors A (8.00i (a) The scalar product O0j) and (9,00i 0Oj) in the xy plane_ determine the following (Enter all angle answers between and 1809.) (b) The angle between the vectors c) The angles and 8 which are respectively the (smallest) angles between the vector and the positive and positive axes (d) The angles and $ which are respectively the (smallest) angles between the vector B and the positive and positive axes



Answers

Vectors $\vec{A}$ and $\vec{B}$ lie in an $x y$ plane. $\vec{A}$ has magnitude 8.00 and angle $130^{\circ} ; \vec{B}$ has components $B_{x}=-7.72$ and $B_{y}=-9.20$ What are the angles between the negative direction of the $y$ axis and $(a)$ the direction of $\vec{A},$ (b) the direction of the product
$\vec{A} \times \vec{B},$ and $(c)$ the direction of $\vec{A} \times(\vec{B}+3.00 \hat{k}) ?$

So a equals four I had Plus Seven J Huh? And B. Eagles. Bye. All right. Minus two Jr The scale of product or the dot product a dot B. It's just, I components multiplied together, plus the J components multiplied. Together we have four times five plus seven times negative, too, in this equals sex uh, 6.0 and then her part. B The angle between the two factors. We can use the relation where the coast sine of the angle is. The dot product bore the norms. So there's just the length of the actors or another dot product. It's six and the norms or just the square root of the squares of the components. We have four squared but seven squared. And this guy, we have bikes where Waas Jews squared. This just equal 0.138 which gives us, if we take the coast inverse to get a tangle of 82.1 degrees

For this question were asked to use Vector A and Vector be given an exercise one point for two. And I wrote those out of the top here. Um, I wrote the I hadn't j had components of both Victor and Victor be for part A. It says find the scaler product of the two vectors and far be wants us to find the magnitude between the two factors. So for part A, the scale of product or also known as the dot product, is gonna be a dot b. So this is equal to the I hacked components of a multiplied by the I had components of be So this is gonna be 4.0 multiplied by 5.0. Plus the J had components of a times the J had components to be So this is gonna be 7.0 times and the negative numbers matter. So this is gonna be times negative 2.0 and it gives us a scaler. That's why they call her scaler product. And a scaler is just a number. Meaning it doesn't have any. I had a J hat values associated with it, so we have 4.0 times 5.0 plus 7.0 times negative, 2.0 or, in other words, 20 minus 14 which comes out to be six course, 6.0 weaken box set in as a solution for a part, B wants us to find the angle between the two. Well, another way of calculating the scaler product not be is also by taking the magnitude of a multiplying it by the magnitude of B and the multiplying that by the coast sine of the angle in between them. We're gonna call that angle theta a B, and that's what we're trying to find a state, A B. But we know that this has to come out to equal 6.0 because that's what we found in part a. So then, if we know the magnitude of A and the magnitude of B, we can solve for the angle fate of a B, get a new page here to do this. So first, let's do the magnitude of a. By definition, the magnitude of A is equal to the sum of the square of both sides. So this is gonna be four squared plus seven squared, so it's gonna be 49 plus 16 all square routed or 65 square root. So Route 65 is the magnitude of a now magnitude of B. This is similar to a do the same thing. This is gonna be five squared, plus negative to square. But a negative squared is a positive. So we can just ignore the negative. The negative sign. So five squared, plus two squared all square root. It comes out to be the square root of 29. Now, if you go back to our previous page, we have nine today times magnitude be times the coastline of data A B is equal to 6.0, right? But now we know magnitude of a magnitude of be so we can solve for data. Maybe we find that data maybe is equal to the inverse co sign, right? We have to take the inverse coastline of both sides to get the angle by itself. So in verse, co sign of the ratio of the scaler product which was 6.0 divided by the magnitude of both, which was square root of 65 times square root of 29 so 6.0 divided by route 65 times, Route 25 or 29. Take the inverse coast side of that ratio and we find out this is equal to 82 0.1 degrees bunkum box set in as our solution to the angle between the two vectors.

In this exercise, we have to find both the scaler product and the angle that these two vectors that are on the screen make with each other. So let me just ah, review very quickly what the scaler product is and how to find the angle from it. So, first of all, a dot be the skate of product is defined as the some of the product of the components that have a X Times be X plus a wide times B y plus z times busy and another way to ride the scale of product is the manager of a times of going to be times, of course, sign between A and the fader, the coastline of the angle fatal between A and B. So this means that the co sign of failure is equal to the scale of product divided by a times Be. So let's apply these concepts to the letters we have. A dot b will be a acts which is four times a B, which is five bus A Why, which is three times view I. She's minus two, so you get 20 minus six, which is equal to 14. This is the scale of product. This is the answer to question a question. We asked us to calculate being a failure. So phase able to the our coal sign off the scaler part of times the magnitude of a times the magnitude of B actually divided by the 19 of eight times, and I'm going to be so. What we need to do is to calculate the magnitude of the in the magnitude would be the magnitude of a is equal to the square root of the X Square e X, where it is 16 plus eight Y square, which is nine. So this is a square root are 25 which is a which is five, and the negative B is equal to the square root of B squared, which is 25 plus view. Observe the X where we just went 25 because of you y squared, which is four. So this is the swear words off 29. So failure is the arc Co sign off 14 divided by five times the square root of 29 and this is equal to 58.7 degrees. Which concludes our question

For this problem we are asked to, for each of the following pairs of vectors calculate the dot product. And on the basis of the results, say whether the angle between the two vectors is acute obtuse or 90 degrees. So for a are vectors are negative to one and 12. So a dot be is going to be negative two times one plus one times two. So that's going to be negative two plus two. So that's going to equal zero. Now, that means then The two factors are perpendicular or their uh or they are 90° to each other For part B We have the vectors to 3 -1. Actually, I won't bother rewriting them. I'll just go directly to the dot product. The dot product of a dot be for part B. It's going to be two times four. So that's going to be eight Plus three times 3. So plus nine Then we have -17. Well, eight plus nine is going to be positive 17 and then we have minus 17. So we end up with zero. So again, the angle is 90 degrees than for part C. A dot be Is going to be one times 3 plus negative two times negative two. So that's going to be double negative. So three plus four then plus five times negative two, so minus 10. Three plus four is going to give us that is seven, 7 -10. Then is going to give us a result of -3. Now, if we have a negative dot product then that indicates that we have an obtuse angle between them.


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