What is the use of the Dot Product of two vectors?

Question

Suppose you have two vectors a and b that you want to take the dot product of, now this is done quite simply by taking each corresponding coordinate of each vector, multiplying them and then adding the result together. At the end of performing our operation we are left with a constant number.

My question therefore is what can we do with this number,why do we calculate it so to speak? I mean it seems almost useless to me compared with the cross product of two vectors (where you end up with an actual vector).

Answer 1

Re: "[the dot product] seems almost useless to me compared with the cross product of two vectors ".

Please see the Wikipedia entry for Dot Product to learn more about the significance of the dot-product, and for graphic displays which help visualize what the dot product signifies (particularly the geometric interpretation). Also, you'll learn more there about how it's used. E.g., Scroll down to "Physics" (in the linked entry) to read some of its uses:

Mechanical work is the dot product of force and displacement vectors.
Magnetic flux is the dot product of the magnetic field and the area vectors.

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You've shared the algebraic definition of the dot product: how to compute it as the sum of the product of corresponding entries in two vectors: essentially, computing $\;\mathbf A \cdot \mathbf B = {\mathbf A}{\mathbf B}^T.\;$

But the dot product also has an equivalent geometric definition:

In Euclidean space, a Euclidean vector is a geometrical object that possesses both a magnitude and a direction. A vector can be pictured as an arrow. Its magnitude is its length, and its direction is the direction the arrow points. The magnitude of a vector A is denoted by $\|\mathbf{A}\|.$ The dot product of two Euclidean vectors A and B is defined by

$$\mathbf A\cdot\mathbf B = \|\mathbf A\|\,\|\mathbf B\|\cos\theta,\quad\text{where $\theta$ is the angle between $A$ and $B.$} \tag{1}$$

With $(1)$, e.g., we see that we can compute (determine) the angle between two vectors, given their coordinates: $$\cos \theta = \frac{\mathbf A\cdot\mathbf B}{\|\mathbf A\|\,\|\mathbf B\|}$$

Answer 2

The original motivation is a geometric one: The dot product can be used for computing the angle $\alpha$ between two vectors $a$ and $b$:

$a\cdot b=|a|\cdot|b|\cdot \cos(\alpha)$.

Note the sign of this expression depends only on the angle's cosine, therefore the dot product is

• $<0$ if the angle is obtuse,
• $>0$ if the angle is acute,
• $=0$ if the $a$ and $b$ are orthogonal.

Another important special case appears when $a=b$: The root of the scalar product of a vector with itself is the length of a vector:

$a\cdot a=|a|\cdot|a|\cdot1=|a|^2$.

There's another interesting application of the dot product, in combination with the cross product: If you have three vectors $a$, $b$ and $c$, they define a parallelepiped, and you can compute its (signed) volume $V$ as follows using the so-called scalar triple product:

$V=(a\times b)\cdot c$

(Note that this is a generalization of $|a\times b|$ being the area of the parallelogram defined by $a$ and $b$.)

Answer 3

Before addressing your question, i want to say that this is a very good question and you are right to expect that the dot product has use/significance.

First, it is important that you think about vectors separate from their coordinates. While it is true that we often represent vectors as a series of coordinates along well-defined axes, this is merely for computational reasons. A vector as an idea "exists" in a space without any predefined coordinate system. I say this because there are two definitions of the dot product, one is coordinate free (i.e. $\mathbf a\cdot\mathbf b = \|\mathbf a\|\,\|\mathbf b\|\cos\theta$) and the other is based on coordinates (i.e. $\mathbf a\cdot\mathbf b = \sum_i{a_i b_i}$). Of these two, it is best to think of the dot product in terms of the former as it does not depend on a coordinate system. (It is relatively easy to show that the latter may be derived from the former, but in that derivation is an implicit assumption that the coordinate system being used to represent the dot product is orthogonal.)

Second, given the coordinate-free definition, the fundamental idea of the dot product is that of projection. By this it gives a single number which indicates the component of a vector in the direction of another vector. Your observation of the dissimilarity between the dot and cross product is correct, however, the dot product is used to produce a vector as well, it just does it component-by-component. Let's suppose that we have a vector $\mathbf v$ represented by its components in a given coordinate system. Let's further suppose that we have an orthonormal basis defined in that same coordinate system as the set of column vectors $\{\mathbf u_1, \mathbf u_2, \ldots, \mathbf u_n\}$. Finally, suppose that we want to represent $\mathbf v$ in this basis as $\mathbf w$. The question is how do we do that? We use the dot product of course! So the first component of $\mathbf w$ would then be $w_1 = \mathbf u_1\cdot \mathbf v$, and the second component would be $w_2 = \mathbf u_2\cdot \mathbf v$ and so on. (Note that because $\|\mathbf u_i\| = 1$, we have $\mathbf u_1\cdot \mathbf v= \|\mathbf v\|\cos\theta_i$.) If we then think of the vector $\mathbf w$ defined as such we have

$$\mathbf w = \left[ \begin{array}{c} w_1 \\ w_2 \\ \vdots \\ w_n \end{array} \right] = \left[ \begin{array}{c} \mathbf u_1\cdot \mathbf v \\ \mathbf u_2\cdot \mathbf v \\ \vdots \\ \mathbf u_n\cdot \mathbf v \end{array} \right] = \left[ \begin{array}{c} \mathbf u_1^T \mathbf v \\ \mathbf u_2^T \mathbf v \\ \vdots \\ \mathbf u_n^T \mathbf v \end{array} \right] = \left[ \begin{array}{c} \mathbf u_1^T \\ \mathbf u_2^T \\ \vdots \\ \mathbf u_n^T \end{array} \right]\mathbf v = \left[ \begin{array}{cccc} \mathbf u_1 & \mathbf u_2 & \cdots &\mathbf u_n \end{array} \right]^T\mathbf v$$

Finally, we conclude that the dot product plays a key role in the transformation of a vector from one basis to another and that the dot product is hidden in the definition of matrix multiplication in that one view of a matrix-vector product is that each element in the product represents a dot product between a row of the left and a column of the right.

I hope this helps.

Answer 4

The dot product is an essential ingredient in matrix product. The product of the two matrices $A$ and $B$ (of compatible sizes, that is, the number of columns of $A$ equals the number of rows of $B$) is a matrix whose $(i, j)$ component is the dot product of the $i$-th row of $A$ and the $j$-th column of $B$.

Among the many applications, consider this simple one.

You have students $s_{1}, \dots, s_{n}$ taking courses $c_{1}, \dots, c_{m}$. Consider the matrix $A$ which has $0$ everywhere, except that the $(i,j)$ coefficient is $1$ if the student $s_{j}$ takes the course $c_{i}$. Now note that the dot product of the $i_{1}$-th row of $A$ with the $i_{2}$-th row gives the number of students that take both $c_{i_{1}}$ and $c_{i_{2}}$. In other words, the matrix $A A^{t}$ has in its $(i, j)$ position the number of students taking both $c_{i}$ and $c_{j}$.

This matrix is of course useful in building a course timetable.

Answer 5

The geometric idea of the dot product has been touched upon, but there is a vast generalization of this product in geometric algebra, the algebra of not only oriented lines (vectors) but planes, volumes, and more (called blades).

In geometric algebra, we have a generalized dot product of a vector $a$ and another blade $B$ denoted $a \cdot B$. This product has a simple geometric interpretation as the part of $B$ orthogonal to the projection of $a$, with magnitude $|a||B|| \cos \theta|$, where $\theta$ is the angle $a$ forms with its projection in $B$.

(Note: as a point of fact, $a \cdot B$ is orthogonal to $a$ also, not just its projection into $B$, but thinking about this leads to some difficulties, while thinking about what's perpendicular to the projection does not.)

If $B$ is a 2-blade (also called a bivector), then you should be able to imagine this directly: if $a$ lies entirely in $B$, then $a \cdot B$ is just the vector perpendicular to $a$ in $B$. If $a$ does not lie entirely in $B$, then it can be decomposed into a tangential part and a normal part. We throw away the normal part, and the previous logic applies for the tangential part.

If $B$ is a 3-blade (a trivector), then in 3d space $a$ must lie in $B$ (for there is no 3d volume that a vector does not help span), and the product $a \cdot B$ is the "Hodge dual", or the plane perpendicular to $a$.

In this light, the dot product of vectors may actually be the most non-intuitive part of this reasoning. When you take the dot product, there's only a scalar left--there's no vector or other higher dimensional object left to be orthogonal to $a$. Again, this is why I emphasize that $a \cdot B$ is the part of $B$ orthogonal to the projection of $a$ onto $B$. When $B$ is a vector, it's clear there is no other vector or anything else that can be orthogonal to the projection of $a$, for $B$ and that projection are parallel, so the result is necessarily just a scalar.

Answer 6

When one calculates A.B, two measurements happen: measurement of how small the angle between them is, and how long A and B are. A.B basically means projection length of A on B, with this length then scaled by the absolute length of B.

One way to think about the interpretation of the dot product is to think how would one maximise or minimise the dot product between two vectors. Let's assume we are trying to maximise the dot product between two vectors that we can modify:

The dot product will be grow larger as the angle between two vector decreases. The dot product A.B will also grow larger as the absolute lengths of A and B increase. This is because as A gets larger, its projected length will be longer, and as B's length gets larger, the scaling of A's projection will grow larger, given that B's absolute length will act as a scaler of A's projection length.

Hence in problems where is desirable to maximise or minimise the size of vectors and minimise the deviation or angle between them, quantifying it using the dot product can be useful.

Answer 7

I don’t see enough plain English answers here, to be honest. One simple example: you can determine in a stealth game whether an object is within a 90 degree line of sight of something, or not.