Understanding Dot Product: Identity and Unit Vectors

This can also be written as \vec A \cdot \hat e. The dot product can be thought of as a measure of the projection of one vector onto another. When the unit vector and non-unit vector are perpendicular, the dot product is 0. When they are parallel, the dot product is equal to the magnitude of the non-unit vector. In summary, the dot product of a unit vector with a non-unit vector is a measure of their projection onto each other and can be simplified to the magnitude of the non-unit vector times the cosine of the angle between them.
  • #1
don_anon25
36
0
What is the dot product of a unit vector (e) with a non-unit vector (A)? Is this some sort of identity?

Thanks,
don_anon25
 
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  • #2
don_anon25 said:
What is the dot product of a unit vector (e) with a non-unit vector (A)? Is this some sort of identity?

Thanks,
don_anon25

Let a be a unit vector and b be another vector. a * b = |a||b|cosθ. Since a is a unit vector, this becomes a * b = |b|cosθ. I believe that without more information, this is as far as it can be simplified.
 
  • #3
don_anon25 said:
What is the dot product of a unit vector (e) with a non-unit vector (A)? Is this some sort of identity?

Thanks,
don_anon25

[tex]
\hat e\cdot\vec A[/tex] is the component of [tex]\vec A[/tex] along the [tex]\hat e [/tex]-direction.
 

FAQ: Understanding Dot Product: Identity and Unit Vectors

What is a dot product?

A dot product, also known as a scalar product, is a mathematical operation that takes two vectors and returns a single scalar value. It is calculated by multiplying the corresponding components of the two vectors and then adding them together.

How is a dot product useful in science?

The dot product has many applications in science, particularly in physics and engineering. It can be used to calculate the work done by a force, determine the angle between two vectors, and find the projection of one vector onto another.

What is the difference between a dot product and a cross product?

A dot product and a cross product are two different types of vector operations. While a dot product results in a scalar value, a cross product returns a vector. Additionally, the dot product measures the similarity between two vectors, while the cross product measures the perpendicularity.

Can a dot product be negative?

Yes, a dot product can be negative. This occurs when the angle between the two vectors is greater than 90 degrees. In this case, the dot product represents the projection of one vector onto the other, which can result in a negative value.

What is the geometric interpretation of a dot product?

The geometric interpretation of a dot product is that it measures the similarity between two vectors. When the dot product is positive, it means the two vectors are pointing in the same direction, and when it is negative, the two vectors are pointing in opposite directions. A dot product of zero indicates that the two vectors are perpendicular to each other.

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