# Proving Parallel Vectors: |u•v| = ||u|| ||v||

• dangish
In summary, the dot product of two vectors, u and v, will be equal to the product of their lengths, ||u|| and ||v||, if and only if they are parallel. This can be proven using the definition of the dot product and a theorem. It is important to be specific when discussing vector multiplication, as there are different types with different properties.
dangish
Question: Show that |u dot v| = ||u|| ||v|| if and only if u and v are parallel.

I know that u and v are parellel if their dot product is 1.

so if |u dot v| = 1 , fair enough, but how can I show that ||u|| ||v|| will also be equal to 1

any help would be greatly appreciated, thanks in advance :)

Do you know the definition of the dot product?

dangish said:
I know that u and v are parellel if their dot product is 1.

This is not true. You must have misunderstood something...

How did you define the dot product?

Not really.. it's just the product of two vectors isn't it?

No, check out the proper definition and then look at your problem again, it should become very clear

Could you perhaps tell me your definiton of the dot product? :D

dangish said:
Not really.. it's just the product of two vectors isn't it?
This is so vague as to be meaningless.

dangish said:
Could you perhaps tell me your definiton of the dot product? :D

Mark44 said:
This is so vague as to be meaningless.

sorry champ

My point is that since there are several ways that vectors can be multiplied, you need to be more specific.

u $\cdot$ v - the dot product, sometimes called the scalar product because it results in a scalare
u X v - the cross product for vectors in R3
av - scalar multiplication of a scalar and a vector

## What is the definition of parallel vectors?

Parallel vectors are two or more vectors that have the same or opposite direction. This means that they either point in the same direction or in the exact opposite direction, and they have the same magnitude or length.

## How do you prove that two vectors are parallel?

To prove that two vectors are parallel, you can use the dot product formula, which states that if the dot product of two vectors is equal to the product of their magnitudes, then the vectors are parallel. In other words, if |u•v| = ||u|| ||v||, then u and v are parallel.

## What is the significance of proving parallel vectors?

Proving parallel vectors is important in various fields of science, such as physics and engineering, as it helps in analyzing and solving problems involving vectors. It also allows us to determine if two lines or objects are parallel, which has practical applications in real-world situations.

## What are some real-life examples of parallel vectors?

Some examples of parallel vectors in real life include two parallel lines, two forces acting in the same direction, and two railway tracks running side by side. In each of these cases, the vectors have the same direction and magnitude, making them parallel.

## Can two vectors be parallel if they have different magnitudes?

Yes, two vectors can still be parallel even if they have different magnitudes. As long as their dot product is equal to the product of their magnitudes, they are considered parallel. This means that the direction of the vectors is the most important factor in determining if they are parallel, not their magnitudes.

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