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## Homework Statement

1)Give a unit vector

a) in the same direction as

**v**=2

**i**+3

**j**

b)Perpendicular to

**v**

2)

a) Find a vector perpendicular to the plane z=2+3x-y

b)Find a vector parallel to the plane

## Homework Equations

## The Attempt at a Solution

1) a)

Method 1:

**u**=

**v**/||

**v**|| so i get

**u**=2/√13

**i**+ 3/√13

**j**

Method 2:

||

**u**||*||

**v**||cos(θ)=2

arccos(2/√13)=56.31*=θ

||

**u**||=1 so

**u**=cos(56.31)

**i**+ sin(56.31)

**j**=2/√13

**i**+ 3/√13

**j**

Method 3:

**u**.

**v**=||

**v**||

(x

**i**+ y

**j**).(2

**i**+3

**j**)=√13

2x + 3y=√13

Here i get stuck. How do i do this and is there any other way?

b)

Method 1:

**u**.

**v**=||

**u**||*||

**v**||cos(θ)=0

1*√13cos(θ)=0

I get stuck here aswell

Method 2: I realized that if i take the angle between

**v**& the x axis I should get the angle between u and the x axis so I can take sin and cos of those angles to get the unit vector perpendicular

**u**=-3/√13

**i**+2/√13

**j**

The reason I'm doing all these ways is because I'm curious of the most efficient and proper way to solve the problem. My class is currently covering dot products and I was expecting it all to be as simple as u.v=(x

**i**+ y

**j**).(2

**i**+3

**j**)=√13 or (x

**i**+ y

**j**).(2

**i**+3

**j**)=0 then I would solve algebraically, but I cant seem to figure out how to do it that way.

2)

a) I know that the normal vector is orthogonal to the pane so

**n**=-3

**i**+

**j**+

**k**

is there another way to do this?

b)

n.v=||

**n**||*||

**v**||cos(θ)=0

(-3

**i**+

**j**+

**k**).(x

**i**+y

**j**+z

**k**)=0

(-3x +y +z)=0

I get the equation of a similar plane but not the one they originally gave me.

Thanks for your help in advance.