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## Homework Statement

From Kleppner and Kolenkow Chapter 1 (Just checking to see if I'm right)

Given vector A=<3, 4, -4>

a) Find a unit vector B that lies in the x-y plane and is perpendicular to A.

b) Find a unit vector C that is perpendicular to both A and B.

c)Show that A is perpendicular to the plane defined by B and C.

## Homework Equations

Cross Product

Dot Product

Knowing how to find a plane from two vectors

## The Attempt at a Solution

a) Using the following system...

3x+4y=0 (using dot product and z=0)

x+y=1 (For it to be a unit vector the sum of the two components must be 1)

The perendicular vector is

##<4, -3, 0>##

and the unit vector after dividing by the magnitude is...

##B=<4/5, -3/5, 0>##

b)Finding the cross product of A and B and dividing by the magnitude...

##C=<\frac{-12,}{5\sqrt{41}}, \frac{-16}{5\sqrt{41}}, \frac{-5}{\sqrt{41}}>##

c)##B \times C=<75, 100, -100>##

Therefore the plane is defined by...

##3x+4y-4z=0##

Therefore A is perpendicular since its components equal the constants that define the plane.

I'm pretty sure I did everything right I just wanted to make sure and see if any of my wording is dumb or if I made a mistake. I don't know how to write the last part formally. Thanks all!