What is the dot product of (b - proj of b onto a) with a?

In summary, the conversation is about a problem involving showing that the vector (orthogonal of b onto a) is orthogonal to vector a. The attempted solution involved manipulating the vectors using the dot product, but the participant got stuck when trying to cancel out vectors and numbers. The other participant suggested writing out "proj of b onto a" in terms of vectors, which could make the problem easier to solve.
  • #1
baokhuyen
9
0

Homework Statement


I get confused with this problems
show that the vector (orth of b onto a) = (b - proj of b onto a) is orthogonal to a.


Homework Equations





The Attempt at a Solution


(b-proj of b onto a) dot a = 0
and I got stuck!
 
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  • #2
Perhaps if you wrote out what "proj of b onto a" was in vector terms it would become easier.
 
  • #3
For example, I say:
(b- a(a.b)/a^2).a=0
(b-(a.b)/a).a=0
a.b-a.((a.b)/a)=0
How can I do next?
 
  • #4
baokhuyen said:
For example, I say:
(b- a(a.b)/a^2).a=0
You are not being careful to distinguish between vectors and numbers. The first "a" of "a(a.b)/a^2" is a vector while "a^2" is a number- the square of the length of a. You are trying to cancel them!

(b-(a.b)/a).a=0
with the result that you get this, which makes no sense! Does "(a.b)/a" mean you are dividing by a vector?

a.b-a.((a.b)/a)=0
How can I do next?
[tex]\left(\vec{b}- \frac{\vec{a}\cdot\vec{b}}{|a|^2}\vec{a}\right)\cdot\vec{a}[/tex]
[tex]\vec{b}\cdot\vec{a}- \frac{\vec{a}\cdot\vec{b}}{|a|^2}(\vec{a}\cdot\vec{a})[/tex]
Now, what is that equal to?
 

1. What is the dot product of two vectors?

The dot product, also known as the 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 adding the results together.

2. How do you calculate the dot product of two vectors?

The dot product of two vectors A and B can be calculated using the formula: A · B = |A||B|cosθ, where |A| and |B| are the magnitudes of the two vectors and θ is the angle between them. This can also be written as the sum of the products of the corresponding components: A · B = A1B1 + A2B2 + ... + AnBn.

3. What is the geometric interpretation of the dot product?

The dot product has a geometric interpretation as well. It is equal to the product of the magnitudes of the two vectors and the cosine of the angle between them. This means that the dot product is positive when the two vectors are pointing in the same direction, negative when they are pointing in opposite directions, and zero when they are perpendicular to each other.

4. What are some real-world applications of the dot product?

The dot product has many real-world applications, such as calculating work done by a force, finding the angle between two vectors, determining if two vectors are perpendicular, and projecting one vector onto another. It is also used in physics, engineering, and computer graphics.

5. Can the dot product be used with vectors in any number of dimensions?

Yes, the dot product can be used with vectors in any number of dimensions. The formula remains the same, summing the products of the corresponding components. However, the geometric interpretation becomes more complex as the number of dimensions increases beyond three.

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