Expressing A for Orthogonal Vectors: Conditions on a,b & c

In summary, the conversation discusses using a 3x3 upper triangular matrix A to ensure that the 3x1 vectors d, e, and f, obtained by multiplying A with vectors a, b, and c, are orthogonal. The elements of A can be expressed in terms of a, b, and c. The condition for finding an orthogonalizing matrix A is dependent on a, b, and c. Additionally, the condition for A to be a diagonal matrix depends on a, b, and d.
  • #1
akanksha331
5
0
Let a,b,c be three 3x1 vectors. Let A be a 3x3 upper triangular matrix which ensures that the 3x1 vectors d,e and f obtained using

[d e f]=A[a b c]

are orthogonal.

a)Express the elements of A in terms of vectors a,b and c.
b)what is the condition on a,b and c which allows us to find an orthogonalizing matrix A?
c)what is the condition on a,b and d which would allow that A be a diagonal matrix?please help me to solve problem
 
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  • #2
Show us your work. No pain, no gain.
 
  • #3
I am sure I can solve the problem . I just want one hint to start the problem.
 
  • #4
Hint:
Read your textbook.
 

1. What is the significance of expressing A for orthogonal vectors?

Expressing A for orthogonal vectors allows us to determine the conditions under which two vectors are perpendicular to each other. This is important in various applications, such as finding the direction of forces in physics or designing efficient algorithms in computer science.

2. What are the conditions for a vector to be orthogonal to another vector?

The conditions for two vectors, a and b, to be orthogonal are:

  • The dot product of a and b must be equal to 0.
  • The length of a and b must be non-zero.
  • The angle between a and b must be 90 degrees.

3. How do you express A for orthogonal vectors in terms of their components?

To express A for orthogonal vectors, we use the formula A = (a1b1 + a2b2 + a3b3), where a1, a2, and a3 are the components of vector a and b1, b2, and b3 are the components of vector b. This formula is derived from the dot product of two vectors.

4. Can three vectors be orthogonal to each other?

Yes, three or more vectors can be orthogonal to each other. In fact, in three-dimensional space, three mutually perpendicular vectors are necessary to define a coordinate system.

5. What is the difference between orthogonal and parallel vectors?

Orthogonal vectors are perpendicular to each other and have a dot product of 0. Parallel vectors, on the other hand, have the same direction and their dot product is equal to the product of their lengths. In other words, orthogonal vectors have an angle of 90 degrees between them, while parallel vectors have an angle of 0 degrees.

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