Find two unit vectors orthogonal to both given vectors

So you can just check that the dot product with (1,1,1) is zero and the length is 1. In summary, The conversation is about finding two unit vectors orthogonal to two given vectors using the cross product. The resulting vectors are <√(14)/14 i, √(14)/7 j, -3√(14)/14 k> and <-√(14)/14 i, -√(14)/7 j, 3√(14)/14 k>, and can be verified by checking their dot product with (1,1,1) and their lengths.
  • #1
C6ZR1
54
0
Hello,

Could someone please review my work and see if it is correct. Thanks :smile:

Homework Statement



Find two unit vectors orthogonal to both given vectors.
i + j + k, 3i + k

Homework Equations





The Attempt at a Solution



So I used cross product and got A x B= i+2j-3k

Then I got magnitude: to be √ (14)

problem asks to rationalize denominator and I eventually get..

lower values: <-√(14)/14 i, -√(14)/7 j, 3√(14)/14 k>

larger values: <√(14)/14 i, √(14)/7 j, -3√(14)/14 k>



I just want to make sure that I did everything correctly.
 
Physics news on Phys.org
  • #2
Looks right to me. You can verify your answers by checking that their inner (dot) products with both given vectors are zero, and that their lengths are 1.

You really only need to check one answer because you know the second answer has to be the negative of the first.
 

What does it mean to find two unit vectors orthogonal to both given vectors?

Finding two unit vectors orthogonal to both given vectors means finding two vectors that are perpendicular to both of the given vectors and have a length of 1.

Why is it important to find unit vectors orthogonal to both given vectors?

It is important to find unit vectors orthogonal to both given vectors because they can be used to represent the basis of a vector space and can simplify calculations and proofs in linear algebra.

How do you find two unit vectors orthogonal to both given vectors?

To find two unit vectors orthogonal to both given vectors, you can use the Gram-Schmidt process or the cross product method. Both methods involve finding a vector that is perpendicular to the given vectors and then normalizing it to have a length of 1.

What is the difference between a unit vector and an orthogonal vector?

A unit vector is a vector with a length of 1, while an orthogonal vector is a vector that is perpendicular to a given vector. A unit vector can be orthogonal to a given vector, but not all orthogonal vectors are unit vectors.

Can there be more than two unit vectors orthogonal to both given vectors?

Yes, there can be infinitely many unit vectors orthogonal to both given vectors. This is because for any given vector, there are infinitely many unit vectors that are perpendicular to it.

Similar threads

Finding Orthogonal Unit Vector to 3 Vectors
    • Calculus and Beyond Homework Help
Replies
1
Views
1K
I cannot make the Gram-Schmidt procedure work
    • Calculus and Beyond Homework Help
Replies
16
Views
1K
One-parameter parametrization of a unit circle in R^n
    • Calculus and Beyond Homework Help
Replies
8
Views
977
Using Inner Product Properties to Solve Vector Problems
    • Calculus and Beyond Homework Help
Replies
4
Views
989
Obtain an equation of the plane in the form ##px+qy+rz=d##
    • Calculus and Beyond Homework Help
Replies
5
Views
184
Two vector operations and simple expressions
    • Calculus and Beyond Homework Help
Replies
6
Views
794
Finding orthogonal unit vector to a plane
    • Calculus and Beyond Homework Help
Replies
2
Views
2K
Derivative of Cosine with unit vector
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
I Non orthogonal basis and the lines of its coordinate grid
    • Linear and Abstract Algebra
Replies
9
Views
198
Unit tangent vector of r(t) = (e^t)(cos t ) i + (e^t)(sin t
    • Calculus and Beyond Homework Help
Replies
3
Views
1K

Hot Threads

Recent Insights

Back
Top