Question

find tangent vector to the graph

Answer 1

\[2x^2 + y^2 = 9 \] at (2,1)

Answer 2

@wio

Answer 3

Differentiate

Answer 4

\[
4x+2yy'=0
\]

Answer 5

\[

4(2)+2(1)y'=0
\]

Answer 6

\[
y'=-4
\]

Answer 7

ok wait y prime = -2x/y

Answer 8

so how do we change it to tangent vector

Answer 9

Just gonna be \(\langle 1,-4\rangle\) I believe.

Answer 10

i mean how, i know how u got y prime how about x prime

Answer 11

I differentiated with respect to \(x\), so \(D_x x=1\)

Answer 12

gotcha thanks and <1,-4> is tangent vector right?

Answer 13

You could use the chain rule and get: \[
4xx'+2yy′=0
\]

Answer 14

ofcourse it is tangent vector as it is the derivative.

Answer 15

\[
8x'+2y'=0
\]Think of it like a dot product: \[

\langle 8,2 \rangle \cdot \langle x',y' \rangle = 0
\]

Answer 16

Remember to get an orthogonal vector to \(\langle a,b \rangle\) you just use \(\langle -b,a \rangle\)

Answer 17

makes sense but i need tangent vector so by this new dot product method how do we approach <1,-4>

Answer 18

Well \[
\langle 8,2\rangle \cdot \langle 1,-4\rangle =0
\]

Answer 19

<8,2> * <x',y'> = 0 ----> <1,-4> ?????

Answer 20

Yeah

Answer 21

Or use \[
2y'=-8x'
\]Let \(x'=1\) and then:\[
2y'=-8\implies y'=-4
\]

Answer 22

how do u know that x prime is 1

Answer 23

Why not?

Answer 24

idk why cant it be -1 or 0

Answer 25

Well it can't be 0 because the 0 vector has no direction

Answer 26

You can use -1

Answer 27

-1 will give us y prime as 4

Answer 28

which is not right

Answer 29

Sure it is, they're the same direction

Answer 30

ohk wait i m confused can u explain me <8,2> * <x',y'> = 0 from this method how to get x prime = 1 and y' = -4

Answer 31

You didn't specify a magnitude. If you did then it would matter, ore than just direction.

Answer 32

Just let \(x'=1\) or any other value you want it to be and solve for \(y'\). There are infinite solutions.

Answer 33

actually my real question is to find directional derivative in direction of tangent vector to this graph

Answer 34

so does it matter if i use any of <1,-4> or <-1, 4>

Answer 35

No

Answer 36

It doesn't matter. Those both point in the same proportional direction.

Answer 37

ok so i will do the dot product of <1,-4> and gradient thanks