Linear Algebra: Dot Product and Orthogonality

In summary, the conversation is about a homework problem that the person is struggling with. They mention not being able to find a reference for solving it and ask for help. The problem involves finding the dot product of two vectors, (3u - 8v) and (3u + 8v), given the lengths of u and v. After receiving a tip to use the linear properties of the dot product, the person was able to solve the problem.
  • #1
mateomy
307
0
This is from my homework, I was moving along nicely until I hit this problem, (there's another just like it right after this). I can't find reference for solving this in the chapter I am looking at. The answer is in the back of the book….-2911. Can someone explain this to me?[tex]
||\mathbf{u}|| = 5\, and\: ||\mathbf{v}|| = 7; \,find\: (3\mathbf{u} - 8\mathbf{v}) \circ (3\mathbf{u} + 8\mathbf{v}).
[/tex]I know that [itex]||\mathbf{u}||[/itex] is just the length of the vector, etc. I just don't know where to go from there.
 
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  • #2
try expanding the dot product using its linear properties, should simplify a heap
 
  • #3
You were right. Solved it first go after that. Thank you.
 

1. What is the dot product in linear algebra?

The dot product, also known as the inner product or scalar product, is a mathematical operation that takes two vectors and produces a scalar quantity. It is calculated by multiplying the corresponding components of the two vectors and then adding them together.

2. How is the dot product used in linear algebra?

The dot product is used to determine the angle between two vectors, as well as the projection of one vector onto another. It is also used to define orthogonality, where two vectors are perpendicular to each other and their dot product is equal to 0.

3. What is orthogonality in linear algebra?

Orthogonality refers to the relationship between two vectors that are perpendicular to each other, or at a right angle. This means that their dot product is equal to 0. In linear algebra, orthogonality is important for solving systems of equations and for finding the best fit line in regression problems.

4. How do you determine if two vectors are orthogonal?

To determine if two vectors are orthogonal, you can calculate their dot product. If the dot product is equal to 0, then the vectors are orthogonal. Another way to determine orthogonality is by checking if the angle between the two vectors is 90 degrees.

5. What are some real-world applications of the dot product and orthogonality in linear algebra?

The dot product and orthogonality have many practical applications in fields such as physics, engineering, and computer science. For example, in physics, the dot product is used to calculate work and energy, while in computer graphics, it is used to determine the angle between two surfaces for lighting and shading. Orthogonality is also used in signal processing and data compression techniques.

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