Perpendicular Unit Vectors in the x-y Plane: Is My Solution Correct?

In summary, the conversation discusses finding a unit vector B perpendicular to vector A in the x-y plane, another unit vector C perpendicular to both A and B, and showing that A is perpendicular to the plane defined by B and C. The solution involves using cross product and dot product, and finding the equation of a plane from two vectors. The solution provided is correct, but there is a mistake in assuming x+y=1 for a unit vector.
  • #1
PhotonSSBM
Insights Author
154
59

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!
 
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  • #2
You can show things are perpendicular by computing the dot product between two vectors.
 
  • #3
PhotonSSBM said:

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)
NO! For it to be a unit vector we must have [itex]\sqrt{x^2+ y^2}= 1[/itex], not x+ y!

The perendicular vector is
##<4, -3, 0>##
and the unit vector after dividing by the magnitude is...
##B=<4/5, -3/5, 0>##
So x+ y is NOT 1!

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!
 
  • #4
HallsofIvy said:
NO! For it to be a unit vector we must have [itex]\sqrt{x^2+ y^2}= 1[/itex], not x+ y!
I see, but since the unit vector I found follows that condition...
##\sqrt{(\frac{4}{5})^2 + (\frac{-3}{5})^2} = 1##
is my solution still right?
 

Related to Perpendicular Unit Vectors in the x-y Plane: Is My Solution Correct?

1. What is a perpendicular unit vector?

A perpendicular unit vector is a vector that is perpendicular, or at a 90 degree angle, to another vector and has a magnitude of 1. This means that it has a length of 1 unit and is used to represent a direction in space.

2. How is a perpendicular unit vector calculated?

A perpendicular unit vector can be calculated by taking the original vector and dividing it by its magnitude. This will result in a vector with the same direction as the original, but with a magnitude of 1.

3. What is the importance of perpendicular unit vectors in physics and mathematics?

Perpendicular unit vectors are important in physics and mathematics because they are used to represent directions in 3-dimensional space. They are also used in vector operations, such as finding the dot product and cross product of two vectors.

4. How are perpendicular unit vectors used in real-world applications?

Perpendicular unit vectors are used in a variety of real-world applications, such as in engineering and navigation. They are used to represent forces, velocities, and directions in 3-dimensional space, making them essential in understanding and solving problems in these fields.

5. Can a vector have more than one perpendicular unit vector?

Yes, a vector can have an infinite number of perpendicular unit vectors. This is because for any given vector, there are an infinite number of possible directions that are perpendicular to it. However, only one of these perpendicular unit vectors will have the same direction as the original vector.

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