# Multivariable Calculus - Dot Products

• engstudent363
In summary, the dot product shows the projection of one vector onto another. It is very useful in physics and maths, and can be used to find the angle between two vectors.
engstudent363
Anyone familiar with dot products of two vectors? What does the dot product show, in other words what is the point of doing a dot product?

Welcome to PF,

I'm assuming that this is a homework assignment, so what is the definition of the scalar product? How to you calculate it? How is it related to the angle between the two vectors?

I'm still confused. Here's an example. I am given a problem X "dot" X and that would give me x1*x1 + x2*x2 + x3*x3 . What is the point of having this information? What does the dot product mean geometrically?

One can also define the dot product thus,

$$\underline{v_1}\cdot\underline{v_2}=\left|\underline{v_1}\right|\left|\underline{v_2}\right|\cos\theta$$

Where $\theta$ is the angle between the two vectors and $\left|\underline{v_1}\right|$, $\left|\underline{v_2}\right|$ are the magnitudes of the respective vectors. Therefore, geometrically the scalar product represent the projection of $\underline{v_1}$ on the unit vector in the direction of $\hat{\underline{v_2}} = \underline{v_2}/\left|\underline{v_2}\right|$. The opposite is also true. For more information see http://mathworld.wolfram.com/DotProduct.html"

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thank you sir. i think i understand it now. By the way, how did you make those cool symbols like for theta or the dot in between v1 and v2

engstudent363 said:
thank you sir. i think i understand it now. By the way, how did you make those cool symbols like for theta or the dot in between v1 and v2
By using [t e x]symbol notation here[/t e x] or [i t e x]symbol notation here[/i t e x].

The dot product has another useful use:

Work = Force•distance

flebbyman said:
The dot product has another useful use:

Work = Force•distance
Well, I certainly wouldn't put it that way. It is common to think of Force as a vector but "distance" is just a number. I would say Work = Force•displacement.

One important application, of which the work formula above is a specific example, is finding the projection of one vector on another. If $\vec{u}$ and $\vec{v}$ are vectors, then the "projection of $\vec{u}$ on $\vec{v}$ is:
$$\frac{\vec{u}\cdot\vec{v}}{||\vec{v}||}$$.

And, of course, the fact that $\vec{u}\cdot\vec{v}= 0$ if and only if $\vec{v}$ and $\vec{u}$ are perpendicular.

In general, $\vec{u}\cdot\vec{v}= ||\vec{u}||||\vec{v}|| cos(\theta)$ where $\theta]$ is the angle between the vectors. "Formally" extending that to vectors with more than three components allows us to define the angle between two vector in higher dimensions.

HallsofIvy said:
Well, I certainly wouldn't put it that way. It is common to think of Force as a vector but "distance" is just a number. I would say Work = Force•displacement.

Oh of course, my bad. The dot product is extremely useful throughout physics and maths, such as in Line Integrals.

In my experience, the dot product tells you more about the geometry than the other way around. What I mean is that often times, although you do things in real life one way (defining vectors as being having distance and angles and stuff and then defining your dot products in terms of these), mathematicians will often turn it around and start with the vectors and then define distance and angles in terms of the dot product.

This is why often times the formulas relating distance and angles to the dot product are sometimes unintuitive. (And why the dot product is a fairly confusing beast at first) In real life, it's useful to think one way. Mathematicians, however, have found that it's sometimes more useful to basically think backwards.

As an example, using the formula $$\underline{v_1}\cdot\underline{v_2}=\left|\underline{v_1}\right|\left|\underline{v_2}\right|\cos\theta$$, you can show that 2 vectors at right angles to each other have a dot product of 0. Mathematicians on the other hand will often define right angles as having a dot product of 0.

Makes some things really nice, let's you easily generalize a lot of things that you find out about vectors... but confuses the heck out engineering and physics students

## What is a dot product in multivariable calculus?

A dot product, also known as an inner product, is a mathematical operation that takes two vectors and produces a scalar value. In multivariable calculus, the dot product is used to find the angle between two vectors and to project one vector onto another.

## 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 up those products. For example, if vector A is (a1, a2, a3) and vector B is (b1, b2, b3), then the dot product A · B = (a1 * b1) + (a2 * b2) + (a3 * b3).

## What is the geometric interpretation of the dot product?

The dot product can be interpreted geometrically as the product of the magnitudes of two vectors and the cosine of the angle between them. This means that the dot product will be larger when the vectors are parallel and smaller when they are perpendicular.

## What is the relationship between the dot product and orthogonality?

The dot product is closely related to orthogonality, or perpendicularity, of two vectors. If the dot product is equal to 0, then the vectors are orthogonal to each other. This means that they are at a 90 degree angle to each other and have no component in the same direction.

## How is the dot product used in multivariable calculus applications?

The dot product has many applications in multivariable calculus, including finding the work done by a force moving an object, determining the rate of change of a function in a given direction, and finding the equation of a plane using a normal vector. It is also used in optimization problems and in calculating surface area and volume of three-dimensional objects.

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