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

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).

Solution 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.

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: $$\cos \theta = \frac{\mathbf A\cdot\mathbf B}{\|\mathbf A\|\,\|\mathbf B\|}$$

Solution 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 $b$.)

Solution 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.