# STUPID Vector qusetion - dont understand dot product rule

• thomas49th
In summary, when you dot two vectors together, the result is the length of the vector between the points that were dot-vectored together.
thomas49th

## Homework Statement

The points A and B have position vectors a = (2,2,1) and b (1,1,-4) respectively relative to an origin O. (im using column notation for shorthand)

Prove that OA is perpendicular to AB

## The Attempt at a Solution

To be perpendicular the angle between the lines is 90

using the dot product rule:

vectors a and b should multiply to give cos 90 (which is 0)

AB = (3,3,-3)-(2,2,1) = (1,1,-4)
so:
(2,2,1).(1,1,-4) = (2,2,-4)

NOW WHAT I DONT UNDERSTAND IS why you add the components i,j,k to get 0. I can see 2 + 2 -4 =0, but why do you do this? Why can you do this

Thanks :)

Simple answer: A dot B is a number (scalar) - not a vector. It's definition is: axbx + ayby + azbz.

thomas49th said:

## Homework Statement

The points A and B have position vectors a = (2,2,1) and b (1,1,-4) respectively relative to an origin O. (im using column notation for shorthand)
You're using row notation. Columns are vertical.
thomas49th said:
Prove that OA is perpendicular to AB
Are you sure you have written the problem correctly? Vectors OA and OB are perpendicular, but OA and AB aren't.
thomas49th said:

## The Attempt at a Solution

To be perpendicular the angle between the lines is 90

using the dot product rule:

vectors a and b should multiply to give cos 90 (which is 0)

AB = (3,3,-3)-(2,2,1) = (1,1,-4)
Where did you get (3, 3, 3)? Vector AB = OB - OA, which is (1, 1, -4) - (2, 2, 1) = (-1, -1, -5).
thomas49th said:
so:
(2,2,1).(1,1,-4) = (2,2,-4)

NOW WHAT I DONT UNDERSTAND IS why you add the components i,j,k to get 0. I can see 2 + 2 -4 =0, but why do you do this? Why can you do this

Thanks :)

Mark44 said:
You're using row notation. Columns are vertical.
It's just ordered-set notation; neither row nor column vectors have commas!

Hi,
You did show that OA is perpendicular to OB. That's good.

I think your question is "Why does the dot-product rule work? Why do you multiply similar components, and then add up the sum?"

Look at the dot-product of A.A: it is simply the Pythagorean Theorem.
If A = (ax, ay, az)
Then A.A = ax2 + ay2 + az2.
So ... the dot-product is a method of determining the length of a vector: Lth (A) = $$\sqrt{A.A}$$

That's just an example to make you feel comfortable with the rule - because it's useful and makes sense in that situation.

Good luck.

BobM

## 1. What is the dot product rule?

The dot product rule is a mathematical formula used to find the scalar quantity resulting from the multiplication of two vectors. It is also known as the scalar product or inner product.

## 2. How is the dot product rule calculated?

The dot product rule is calculated by multiplying the corresponding components of two vectors and then adding the results together. For example, if we have two vectors A and B with components a1, a2, a3 and b1, b2, b3 respectively, the dot product would be calculated as a1*b1 + a2*b2 + a3*b3.

## 3. What is the purpose of the dot product rule?

The dot product rule is commonly used in physics and engineering to calculate the work done by a force, the angle between two vectors, and projections of one vector onto another. It is also used in linear algebra and vector calculus to solve problems involving vectors.

## 4. Can you explain how the dot product rule is related to the angle between two vectors?

The dot product rule is used to calculate 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. So, by rearranging the equation, we can find the angle between two vectors as cos⁡(θ) = (A⋅B)/(∥A∥∥B∥), where A and B are the two vectors and θ is the angle between them.

## 5. Are there any limitations to using the dot product rule?

While the dot product rule is a useful tool in solving problems involving vectors, it has some limitations. It can only be used for vectors in Euclidean space, meaning they have a defined magnitude and direction. Additionally, the dot product rule only works for two-dimensional and three-dimensional vectors, and not for higher dimensions.

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