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Enter the values of the two given vectors to find the dot product.

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This dot product calculator helps to calculate the dot (inner) product of two vectors using coordinates, points, or magnitude and angle.

It provides a step-by-step vector multiplication solution using the formula of the dot product.

You can find the product (multiplication) of the 2-dimensional and 3-dimensional vectors with this online dot product solver.

In mathematics, the dot product is a way to multiply the lengths of the vectors.

The result of this multiplication is a scalar value. Therefore it is also known as a scalar product.

Geometrically, it is the product of the Euclidean magnitudes and the cosine of the angle between the vectors.

The dot product tells you how much of the first arrow "line up" with the second arrow:

- If both the arrows point in the same direction, then it means that the dot product is a positive number
- If the arrows are pointing in the opposite direction, then the result of the dot product is a negative number
- When the arrows are perpendicular (or orthogonal), the dot product is zero

There are three methods to find the dot product of the vectors, which are:

Using The Vector Components Formula (Vector Coordinates & Points)

**\(\overrightarrow u = (u_1, u _2, u_3),\ and\ \overrightarrow v = (v_1, v_2, v_3)\)**

**\(\vec u\cdot \vec v= (u_{1}\hat{i} + u_{2}\hat{j} + u_{3}\hat{k})\ \cdot (v_{1}\hat{i} + v_{2}\hat{j} + v_{3}\hat{k})\)**

**\(\ (u_{1}v_{1})(\hat{i}.\hat{i}) + (u_{1}v_{2})(\hat{i}.\hat{j}) + (u_{1}v_{3})(\hat{i}.\hat{k})\ +\)**

**\(\ (u_{2}v_{1})(\hat{j}.\hat{i}) + (u_{2}v_{2})(\hat{j}.\hat{j}) + (u_{2}v_{3})(\hat{j}.\hat{k})\ +\)**

**\(\ (u_{3}v_{1})(\hat{k}.\hat{i}) + (u_{3}v_{2})(\hat{k}.\hat{j}) + (u_{3}v_{3})(\hat{k}.\hat{k})\)**

\(\hat{i}.\hat{j} = \hat{i}.\hat{k} =\hat{k}.\hat{j}\) = cos 90 = 0, Thats why these vectors are orthogonal

\(\hat{i}.\hat{i} = \hat{k}.\hat{k} =\hat{j}.\hat{j}\) = cos o = 1, (Codirectional Vectors)

**\(\vec u.\vec v = (u_1v_1\ +\ u _2v_2\ +\ u_3v_3)\)**

This is the most common method when the two vectors are in cartesian coordinates. The dot product of two vectors is based on the projection of one vector onto another and depends on the magnitude of both vectors. It is beneficial when you need to measure the angle between two vectors and want to find the position of a vector relative to the coordinate axes.

**\(\vec{u}\cdot\vec{v}=|\vec{u}||\vec{v}|\cos(\phi) = |\vec{v}|\text{scalarproj}_\vec{v}, (\vec{u})=|\vec{u}|\text{scalarproj}_\vec{u}, (\vec{v})=\)**

**\(\cos(\phi) = \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| \, |\vec{v}|}\)**

- When the dot product of two vectors gives a positive value then the angle between the vectors is considered acute
- But if it is negative then the angle is obtuse
- If the result of the dot product is zero then the vectors are perpendicular to each other

Calculate the dot product of vectors u = (3, 4, 1) and v = (1, 4, -2) and also find the angle between the vectors

**Solution:**

**Identify components:**

u_x = 3, u_y = 4, u_z = 1; v_x = 1, v_y = 4, v_z = -2

By adding values in the formula

\(\ (3\ \ * \ 1) + (4\ \ * \ 4) + (1\ \ * \ 4)\) = 3 +16 + (-2)

3 +16 + (-2) = 17

The dot product of vectors u and v is 17

Now to find the angle, we need to get the magnitude of both vectors, So

**The magnitude of vector**

\(\vec u\):

\(\ |\vec{u}| = \sqrt{(3)^{2} + (4)^{2} + (1)^{2}}\)

\(\ |\vec{u}| = \sqrt{9+ 16 + 1}\)

\(\ |\vec{u}| = \sqrt{26}\)

\(\ |\vec{a}| = \ 5.09\)

**The magnitude of vector**

\(\vec v\):

\(\ |\vec{v}| = \sqrt{(1)^{2} + (4)^{2} + (-2)^{2}}\)

\(\ |\vec{v}| = \sqrt{2+ 16 + 4}\)

\(\ |\vec{v}| = \sqrt{22}\)

\(\ |\vec{v}| = \ 4.69\) (rounded to two decimal places)

**Find the angle:**

\(\theta =\ cos^{-1} \frac{\vec{u} \cdot \vec{v}}{|\vec{u}| \, |\vec{v}|}\)

\(\theta =\ cos^{-1} \frac{17}{\ 5.09\ \ * \ 4.69}\)

\(\theta =\ cos^{-1} (0.7121)\) (rounded to three decimals)

\(\theta =\ 97.53\ deg\) ≈ 67.4 degrees

**\(\vec{u}\cdot\vec{v}=|\vec{u}||\vec{v}|\cos(\phi)\)**

**Where:**

- \(\vec{u}\cdot\vec{v}\) represents the dot product of vectors
- \(\|\vec{u}| \ and\ |\vec{v}|\) are the magnitudes of the vectors \(\vec{u}\ and\ \vec{v}\)
- \(\phi\) is the angle between \(\vec{u}\ and\ \vec{v}\)

Suppose you have two vectors \(\vec u\ and\ \vec v\) with magnitudes \(|\vec u| =\ 5, |\vec v| =\ 7\), and the angle between the vectors is 45 degrees. Calculate the dot product of vectors.

**Solution: **

**Given that:**

- Magnitude of \(\ |\vec u| =\ 5\)
- Magnitude of \(\ |\vec v| =\ 7\)
- The angle (φ) = 45°

**Put values in the dot product formula:**

\(\vec u\cdot \vec v\) = (5) (7) cos(45°) ≈ 24.74 (rounded to two decimal places).

The online calculator for dot products is an instant solution for making vector scalar product calculations.

These are the steps that you only need to follow to calculate the vector’s dot product:

**Step #1:**Enter vector coordinates/components, or magnitudes & angles (if available)**Step #2:**Click on “Calculate” to get the result from the dot product vector calculator

Here are the five properties of the dot product:

- Commutative: (a.b = b.a)
- Distributive Over Vector: (a.(b + c) = a.b + a.c)
- Bilinear: (ra.b = r(a.b) and a.(rb) = r(a.b))
- Scalar Multiplication: (ca.b = c(a.b))
- Not Associative: a.(b.c) ≠ (a.b).c)

The dot product of two vectors plays a crucial role in physics because it allows us to quantify specific relationships between vectors that represent physical quantities like displacement, force, and velocity. These relationships are important because they involve the direction of the vectors, not just their magnitude. Thankfully, using a scalar product calculator can simplify the physics problems that involve the multiplication of vectors.

- Installing a solar panel on a roof
- Finding projections
- Work done by a force
- Testing for orthogonality
- Image and signal processing

- The result of a dot product is a scalar quantity and it only represents the magnitude
- The cross-product results in a vector quantity and it indicates both magnitude and direction

From the authorized source of Wikipedia: Dot product.

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