How Do Dot Products Reflect Vector Projections?

In summary, the dot product of two 2-dimensional vectors can be calculated using two different equations: (ax * bx) + (ay * by) and a*bCos(θ). This is because the dot product involves finding the projection of one vector onto another, which can be thought of as the scalar component of one vector in the direction of the other. This concept can be further explained and understood through the article on Better Explained.
  • #1
Pochen Liu
52
2
I know that a dot product of 2, 2 dimension vectors a, b =

(ax * bx) + (ay * by)

but it also is equal to

a*bCos(θ)

because of "projections". That we are multiplying a vector by the 'scalar' property of the other vector which confuses me because that projection is in the direction of the other vector, therefore no scalar, so I cannot see how these two produce the same outcome.

What am I missing intuitively?
 
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Related to How Do Dot Products Reflect Vector Projections?

1. What is the intuition behind the dot product?

The dot product is a mathematical operation that takes two vectors and produces a scalar value. It represents the projection of one vector onto another, giving a measure of how much the two vectors align with each other. It can also be thought of as a measure of similarity between two vectors.

2. How is the dot product calculated?

The dot product is calculated by taking the product of the corresponding components of two vectors and then summing them up. For example, if we have two vectors A = [a1, a2, a3] and B = [b1, b2, b3], the dot product would be calculated as A · B = (a1 * b1) + (a2 * b2) + (a3 * b3).

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

The dot product has many applications in physics, engineering, and computer graphics. It is used to calculate work and energy in physics, find the angle between two vectors, and determine the similarity of documents in natural language processing. It is also used in computer graphics to calculate lighting and shading effects.

4. How does the dot product relate to vector projection?

The dot product can be thought of as a measure of vector projection. When we take the dot product of two vectors, we are essentially finding the length of the projection of one vector onto the other. This can help us understand the relationship between the two vectors and how much they align with each other.

5. Can the dot product be negative?

Yes, the dot product can be negative. This occurs when the angle between the two vectors is greater than 90 degrees, meaning they are pointing in opposite directions. In this case, the dot product represents the amount of "anti-alignment" between the two vectors.

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