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Point) Find the directional dervalive or f(T,y, =) Ty | :2 at ine point (1,2,3) tne directon ot vector maklng an angle of 3 wlth Vf(1,2,3)....

Question

Point) Find the directional dervalive or f(T,y, =) Ty | :2 at ine point (1,2,3) tne directon ot vector maklng an angle of 3 wlth Vf(1,2,3).

point) Find the directional dervalive or f(T,y, =) Ty | :2 at ine point (1,2,3) tne directon ot vector maklng an angle of 3 wlth Vf(1,2,3).



Answers

Use vectors to find the point that lies two-thirds of the way from $P$ to $Q$. $$P(4,3,0), Q(1,-3,3)$$

We're going to use vectors to find the point Um that lies 2/3 of the way from p. d. q. So using these two points, we're going to find the component vector between them. Well 6 -1 is five, 8 -2 is six And 2 -5 is -3. And we want to find the magnitude using this formula. So when we do that we get 25 plus 36 that's nine, that's gonna end up giving us 70. So route 70 is going to be magnitude. So then dividing that if we have one over root 70 times are vector. This gives us the unit factor that goes in the direction from P to Q. Then if we want it to be two thirds of the way, we just need to multiply this by a magnitude of two thirds. It's not our final results.

The question. We are required to find the direction consigns of victor. A. B. Well the coordinates of point A. R. One comma two comma minus three and that of the ar minus one comma minus two comma one supposed to fall. Let us find the representation for victor. Maybe this will be equal to -1 -1 icap plus -2 -2. Jacob plus one plus 3 Giga On simplification. Victor comes out to be -2 cap minus four. Jacob plus four K. Cap magnitude of victory. Be will be under route to square plus four square plus four square. Let us simplify further four plus 16 plus 16. This will be called on the road 36 which is equal to six. So we can say that therefore that direction who science worthy minus two divided by six, comma minus four divided by six comma four divided by six. We can simplify it further as minus one by three comma minus two by three comma. To buy three. This will become the final answer.

Okay. We've been giving these two vectors on. We'd like to find the angle between them. We're gonna be very simple. Formula costs off. Our angle is equal to dot Pericles rt vectors will call them you on DFI over the length of you times by the length off. The so welcome these three individual things on their back into an equation. So you don t is gonna be for two minus 333 times 2 to 3. Should every three times through three. Well, over three were minus. Times a minus is gonna plus on. Then we'll have a to 23 which is six. So what? How? Three times six, which is 18 on the second half, will be minus three times to B minus six. So we end up as a doctorate 12 moving on the length of you. That's gonna be the square root off. You don't put itself minus three, really? Three times by, um, minus three. Re three is due to Drew. Dude, in my three route three 27. So we have 27 here on that. Uh, minus three times minus three is gonna plus nine, So we'll have 27 plus nine, which is 36 which you should know Square is to six. Next up. We need to find the length off. The so that would be on TV. Don't protect itself. So we'll have to route three times by teary three. We should give us 12 on. Then we'll have to do is to which is for four plus 12. 16 And you should know the square 16 is four. Okay, So putting factor equation. We want our cause near the sites that will have caused to minus one 12 on top on six times for on the bottom, which is 24 which means we actually have caused to the minus one off half, which you should know. What if you don't put into calculator is exactly 60 degrees

Yeah. So for this problem we're gonna want to use some important formulas known as the direction coastlines. So understanding the definition of the dot product, we can solve for alpha, beta and gamma. And then we have V. The vector which has components V X V Y. Yeah V X V Y V. Z. In this case what that is were given um six three negative two. Then the last thing we need to find is the magnitude. So the magnitude of the vector is equal to the square root of six square. So that's 36 plus nine plus four. So it's gonna give us seven. So knowing this, knowing this and knowing these formulas, we can calculate the answers. So we'll have alpha, beta and gamma. Alpha is going to equal 31 degrees. Beta is going to equal 65 degrees And gamma is going to equal 107°.. And then in terms of the direction co signs, um that would just be by finding this ratio right here.


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