# Simple dot product [inner product] (verification)

• seto6
In summary, a dot product is a mathematical operation that takes two vectors and produces a scalar value by multiplying their corresponding components and adding them together. It is useful for determining the angle between vectors, calculating projections and work, and determining if vectors are orthogonal or parallel. The dot product is calculated by multiplying and adding the corresponding components of the vectors. There is also a relationship between the dot product and the cosine of the angle between two vectors, and it is used in vector projection by finding the component of one vector in the direction of another vector.
seto6

know dot product

## The Attempt at a Solution

PART A

PART B
not sure what's it asking for

help would be great

Well, what you posted is the answer to the B part actually Notice the "for all v" part in A. Specific example of u,v does the trick for B, not for A. Hint for A: (u-w)*v = 0 for all v. What does it say about u - w?

## 1. What is a dot product?

A dot product, also known as an inner product, is a mathematical operation that takes two vectors and produces a scalar value. It is calculated by multiplying the corresponding components of the vectors and then adding all of the products together.

## 2. Why is a dot product useful?

A dot product is useful because it can tell us the angle between two vectors, whether they are orthogonal (perpendicular) or parallel, and it can also be used in calculations for projections and work.

## 3. How is a dot product calculated?

To calculate a dot product, the corresponding components of the two vectors are multiplied and then added together. For example, if vector a = [2, 3] and vector b = [4, 5], the dot product would be calculated as (2*4) + (3*5) = 23.

## 4. What is the relationship between the dot product and the cosine of the angle between two vectors?

The dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them. This relationship is defined by the dot product formula: a · b = |a| * |b| * cos(θ).

## 5. How is the dot product used in vector projection?

In vector projection, the dot product is used to find the component of one vector that lies in the direction of another vector. This can be calculated by dividing the dot product of the two vectors by the magnitude of the second vector.

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