How To Find The Dot Product Of Two Vectors

Dot Product

The dot product is ane way of multiplying ii or more than vectors. The resultant of the dot product of vectors is a scalar quantity. Thus, the dot product is also known as a scalar product. Algebraically, it is the sum of the products of the corresponding entries of two sequences of numbers. Geometrically, it is the product of the Euclidean magnitude of two vectors and the cosine of the bending between them. The dot product of vectors finds various applications in geometry, mechanics, engineering, and astronomy. Permit us discuss the dot production in item in the upcoming sections.

1.	What is Dot Product of Ii Vectors?
2.	Dot Product Formula
3.	Geometrical Significant of Dot Product
4.	Matrix Representation of Dot Product
five.	Backdrop of Dot Product
6.	Dot Product of Unit Vectors
seven.	Application of Dot Product
8.	FAQs on Dot Production

What is Dot Product of Two Vectors?

Dot Product of vectors is equal to the product of the magnitudes of the two vectors, and the cosine of the angle between the ii vectors. The resultant of the dot product of two vectors lie in the aforementioned airplane of the two vectors. The dot product may be a positive existent number or a negative existent number.

dot product of two vectors

Dot Product Definition

In vector algebra, if ii vectors are given as: a= [\(a_1\),\(a_2\),\(a_3\),\(a_4\),….,\(a_n\)] and b = [\(b_1\),\(b_2\),\(b_3\),\(b_4\),….,\(b_n\)]
then their dot production is given by:
a.b = \(a_1 b_1\)+\(a_2 b_2\)+\(a_3 b_3\)+……….\(a_n b_n\)

\(\overrightarrow a. \overrightarrow b = \sum_{i=1}^{n} a_i b_i\)

Dot Product Formula for Vectors

Allow a and b be two non-aught vectors, and θ exist the included angle of the vectors. Then the scalar product or dot product is denoted past a.b, which is defined as:

\(\overrightarrow a. \overrightarrow b\) = \(|\overrightarrow a||\overrightarrow b|\) cos θ.

Here, \(|\overrightarrow a|\) is the magnitude of \(\overrightarrow a\),

\(|\overrightarrow b|\) is the magnitude of \(\overrightarrow b\), and θ is the bending between them.

Note: θ is not divers if either \(\overrightarrow a\) = 0 or \(\overrightarrow b\) = 0.

Geometrical Meaning of Dot Product

The dot product of two vectors is synthetic past taking the component of 1 vector in the direction of the other and multiplying it with the magnitude of the other vector. To empathize the vector dot product, we start need to know how to find the magnitude of 2 vectors, and the angle betwixt 2 vectors to detect the projection of 1 vector over another vector.

Magnitude of A Vector

A vector represents a direction and a magnitude. The magnitude of a vector is the foursquare root of the sum of the squares of the individual constituents of the vector. The magnitude of a vector is a positive quantity. For a vector \(\overrightarrow a = a_1x + a_2y + a_3z\), the magnitude is |a| and is given past the formula, \(|\overrightarrow a| = \sqrt{a_1^ii + a_2^two +a_3^2}\)

Projection of a Vector

The dot production is useful for finding the component of ane vector in the direction of the other. The vector project of ane vector over some other vector is the length of the shadow of the given vector over another vector. It is obtained by multiplying the magnitude of the given vectors with the cosecant of the bending betwixt the two vectors. The resultant of a vector projection formula is a scalar value.

geometrical meaning of dot product of vectors

Let OA = \(\overrightarrow a\), OB = \(\overrightarrow b\), be the two vectors and θ exist the angle betwixt \(\overrightarrow a\) and \(\overrightarrow b\). Describe AL perpendicular to OB.

From the correct triangle OAL , cos θ = OL/OA

OL = OA cos θ = \(|\overrightarrow a|\) cos θ

OL is the vector projection of a on b.

\(\overrightarrow a. \overrightarrow b\) = \(|\overrightarrow a||\overrightarrow b|\) cos θ = \(|\overrightarrow b|\) OL

= \(|\overrightarrow b|\) (projection of \(\overrightarrow a\) on \(\overrightarrow b\))

Thus, projection of \(\overrightarrow a\) on \(\overrightarrow b = \dfrac{\overrightarrow a. \overrightarrow b}{|\overrightarrow b|}\)

Similarly, the vector project of \(\overrightarrow b\) on \(\overrightarrow a = \dfrac{\overrightarrow a. \overrightarrow b}{|\overrightarrow a|}\)

Angle Betwixt Two Vectors Using Dot Product

The angle between two vectors is calculated as the cosine of the angle between the two vectors. The cosine of the bending between two vectors is equal to the sum of the product of the individual constituents of the 2 vectors, divided past the product of the magnitude of the two vectors. The formula for the angle between the ii vectors is equally follows.

\(cos\theta = \dfrac{\overrightarrow a.\overrightarrow b}{|a|.|b|}\)

\(cos\theta = \dfrac{a_1.b_1 + a_2.b_2 +a_3.b_3}{\sqrt{a_1^2 + a_2^2 +a_3^3}.\sqrt{b_1^ii + b_2^2 + b_3^2}}\)

Working Rule to Observe The Dot Product of 2 Vectors

If the two vectors are expressed in terms of unit of measurement vectors, i, j, yard, along the 10, y, z axes, then the scalar production is obtained every bit follows:

If \(\overrightarrow a = a_1\hat i + a_2 \chapeau j + a_3 \lid k\) and \(\overrightarrow b = b_1 \hat i + b_2 \hat j + b_3\hat one thousand\), then

\(\overrightarrow a. \overrightarrow b\) = \((a_1 \hat i + a_2 \hat j + a_3 \hat yard)(b_1 \hat i + b_2 \chapeau j + b_3 \lid k)\)

= \((a_1b_1) (\hat i. \lid i) + (a_1b_2) (\lid i.\hat j)+ (a_1b_3) (\chapeau i. \hat g) + \\(a_2b_1) (\hat j. \hat i) + (a_2b_2)(\hat j. \hat j) + (a_2b_3 (\hat j. \hat k) + \\(a_3b_1)(\hat k. \hat i) + (a_3b_2)(\hat k. \hat j) + (a_3b_3)(\hat 1000. \hat one thousand)\)

\(\overrightarrow a. \overrightarrow b\) = \(a_1b_1\) + \(a_2b_2\)+ \(a_3b_3\)

Matrix Representation of Dot Product

Information technology is easy to compute the dot product of vectors if the vectors are represented every bit row or column matrices. The transpose matrix of the first vector is obtained equally a row matrix. Matrix multiplication is done. The row matrix and cavalcade matrix are multiplied to become the sum of the product of the corresponding components of the two vectors.

Properties of Dot Production

The following are the properties of the dot product of vectors.

Commutative property
Distributive property
Natural property
General properties
Vector identities

Commutative property of Dot Product:

With the usual definition, \(\overrightarrow a\). \(\overrightarrow b\) = \(\overrightarrow b\) . \(\overrightarrow a\) , we have \(|\overrightarrow a||\overrightarrow b|\) cos θ = \(|\overrightarrow b||\overrightarrow a|\) cos θ

Distributivity of Dot Product

Let a, b, and c exist any iii vectors, so the scalar product is distributive over addition and subtraction. This property can exist extended to any number of vectors.

\(\overrightarrow a. (\overrightarrow b+\overrightarrow c) = \overrightarrow a. \overrightarrow b + \overrightarrow a. \overrightarrow c\)
\((\overrightarrow a+\overrightarrow b). \overrightarrow c = \overrightarrow a. \overrightarrow c+ \overrightarrow b. \overrightarrow c\)
\(\overrightarrow a. (\overrightarrow b - \overrightarrow c) = \overrightarrow a. \overrightarrow b - \overrightarrow a. \overrightarrow c\)
\((\overrightarrow a -\overrightarrow b). \overrightarrow c = \overrightarrow a. \overrightarrow c - \overrightarrow b. \overrightarrow c\)

Nature of Dot Product

We know that 0 ≤ θ ≤ π.
If θ = 0 and then a.b = ab [Two vectors are parallel in the same management ⇒ θ = 0 ] .
If θ = π , a.b = -ab [Two vectors are parallel in the opposite direction ⇒ θ = π.].
If θ = π/two, then a.b = 0 [Two vectors are⇒ θ = π/2]
If 0 < θ < π/2, and so cosθ is positive and hence a.b is positive.
If π/ii < θ < π then cosθ is negative and hence a.b is negative.

Other Properties of Dot Product

Allow a and b be any two vectors, and λ be whatsoever scalar. And then (λ\(\overrightarrow a) . \overrightarrow b\) = λ (\(\overrightarrow a) \overrightarrow b\)
For any two scalars λ and μ, λ\(\overrightarrow a\) . μ \(\overrightarrow b\) = (λμ\(\overrightarrow a)). \overrightarrow b\) = \(\overrightarrow a\). (λμ \(\overrightarrow b\))
The length of a vector is the square root of the dot product of the vector by itself. \(\overrightarrow a\) = \(\sqrt{\overrightarrow a . \overrightarrow a}\)
\(\overrightarrow a. \overrightarrow a\) = \(|\overrightarrow a\)|ⁱⁱ= (\(\overrightarrow a\))²= \(\overrightarrow a\)ⁱⁱ
For any two vectors a and b, \(|\overrightarrow a + \overrightarrow b|\) ≤ |\((\overrightarrow a\)| + |(\(\overrightarrow b|\)

Vector Identities

(\(\overrightarrow a + \overrightarrow b\)) ²= \(|\overrightarrow a\)|²+ \(|\overrightarrow b|\)²+ 2 \((\overrightarrow a.\overrightarrow b)\)
(\(\overrightarrow a - \overrightarrow b\)) ²= \(|\overrightarrow a\)|²+ \(|\overrightarrow b|\)^two- 2 \((\overrightarrow a.\overrightarrow b)\)
\((\overrightarrow a + \overrightarrow b). (\overrightarrow a - \overrightarrow b) = |\overrightarrow a\)|²- \(|\overrightarrow b|\)²≤ \(|\overrightarrow a\)| + \(|\overrightarrow b|\)

Dot Product of Unit Vectors

The dot product of the unit vector is studied by taking the unit vectors \(\chapeau i\) along the ten-axis, \(\hat j\) along the y-centrality, and \(\hat k\) along the z-axis respectively. The dot product of unit vectors \(\lid i\), \(\hat j\), \(\lid k\) follows like rules as the dot product of vectors. The angle between the same vectors is equal to 0º, and hence their dot product is equal to 1. And the angle between two perpendicular vectors is 90º, and their dot production is equal to 0.

\(\hat i.\hat i\) = \(\hat j.\chapeau j\) = \(\chapeau k.\lid k\)= 1

\(\hat i.\hat j\) = \(\hat j.\lid k\) = \(\hat k.\hat i\)= 0

Applications of Dot Product

The application of the scalar production is the calculation of piece of work. The product of the force applied and the deportation is called the work. If strength is exerted at an angle θ to the deportation, the work done is given equally the dot product of forcefulness and displacement equally W = f d cos θ. The dot product is likewise used to exam if ii vectors are orthogonal or not. \(\overrightarrow a. \overrightarrow b\) = \(|\overrightarrow a||\overrightarrow b|\) cos 90º ⇒ \(\overrightarrow a. \overrightarrow b\) = 0

Important Notes on Dot Product

The dot production or the scalar product is a way to multiply two vectors.
Geometrically, the dot product is the product of the length of the vectors with the cosine angle between them. \(\overrightarrow a. \overrightarrow b\) = |a|b| cos θ
It is a scalar quantity having no direction. It is easily computed from the sum of the production of the components of the two vectors.
If \(\overrightarrow a\) = \(a_1\) i + \(a_2\) j + \(a_3\) k and \(\overrightarrow b\)= \(b_1\) i + \(b_2\) j + \(b_3\) grand, so \(\overrightarrow a. \overrightarrow b = a_1b_1 + a_2b_2+ a_3b_3\)

☛ Also Check:

Cantankerous Product
Add-on of Vectors
Production of Vectors
Types of Vectors

Dot Product of Vectors Examples

Example 1: Find the dot production of two vectors having magnitudes of half dozen units and 7 units, and the bending between the vectors is threescore°.

Solution:

The magnitudes of the two vectors are |a| = 6, |b| = 7, and the angle betwixt the vectors is θ = 60°

The dot product of the two vectors is:

a.b = |a|.|b|.Cosθ

= (6).(seven).Cos60°

= (6).(seven).(ane/2)

= (3).(7)

= 21

Answer: a.b = 21
Case 2. Discover the bending between the 2 vectors 2i + 3i + thou, and 5i -2j + 3k.

Solution:

The two given vectors are:

\(\overrightarrow a\) = 2i + 3i + k, and \(\overrightarrow b\) = 5i -2j + 3k

|a| = \(\sqrt{ii^2 + 3^ii + 1^two}\) = \(\sqrt{4 + 9 + ane}\) = \(\sqrt{14}\)

|b| = \(\sqrt{5^ii + (-2)^2 + 3^2}\) = \(\sqrt{25 + 4 + ix}\) = \(\sqrt{38}\)

Using the dot product we have \(\overrightarrow a.\overrightarrow b\) = 2.(five) + 3.(-two) + 1.(three) = x - 6 + 3 = 7

Cosθ = \( \dfrac{a.b}{|a|.|b|}\)

= \(\dfrac{7}{\sqrt{xiv}.\sqrt{38}}\)

= \(\dfrac{7}{ii.\sqrt{7 \times xix}}\)

= \(\dfrac{7}{two \sqrt {133}}\)

θ = Cos^-1\(\dfrac{7}{2 \sqrt {133}}\)

θ = Cos^-10.304 = 72.3°

Answer: Therefore the angle between the vectors is 72.3°
Case 3. Test whether vectors a = 3i + 2j -k and b = i - 2j - k are orthogonal.

Solution:

To check if two vectors are perpendicular, we compute the dot production and find if the result is 0.

Given: a = 3i + 2j -k and b = i - 2j - 1000.

The dot product is computed as \(\overrightarrow a.\overrightarrow b = a_1b_1 + a_2b_2+ a_3b_3\)

= iii(1) + two(-2) +(-1)(-i)

= iii - 4 +1

= 0

Reply: The dot product proves that the vectors a and b are orthogonal.

go to slidego to slidego to slide

Breakdown tough concepts through elementary visuals.

Math volition no longer be a tough field of study, specially when you understand the concepts through visualizations.

Book a Free Trial Form

Practice Questions on Dot Product

go to slidego to slide

FAQs on Dot Product

What is the Dot Product of Two Vectors?

The dot production of two vectors has two definitions. Algebraically the dot production of two vectors is equal to the sum of the products of the individual components of the two vectors. a.b = \(a_1b_1\) + \(a_2b_2\)+ \(a_3b_3\). Geometrically the dot product of two vectors is the product of the magnitude of the vectors and the cosine of the angle between the 2 vectors. ( \(\overrightarrow a. \overrightarrow b\) = \(|\overrightarrow a||\overrightarrow b|\) cos θ). The resultant of the dot product of vectors is a scalar value.

What is the Dot Product of Two Parallel Vectors?

The dot product of two parallel vectors is equal to the production of the magnitude of the two vectors. For 2 parallel vectors, the angle between the vectors is 0°, and Cos0°= ane. Hence for 2 parallel vectors a and b we have \(\overrightarrow a. \overrightarrow b\) = \(|\overrightarrow a||\overrightarrow b|\) cos 0° = |a|.|b|.1 = |a|.|b|.

What is the Deviation Between Dot Product and Cross Product?

The dot product is a scalar product and the cross product is the vector product. The dot product of two vectors is a.b = |a|.|b|Cosθ and the cross product of ii vectors is equal to a × b = |a|.|b| Sinθ. The resultant of the dot production of 2 vectors lie in the aforementioned plane as the two vectors, whereas the resultant of the cantankerous production lies in a airplane perpendicular to the plane spanning the ii vectors.

What is Dot Product Formula?

The dot production formula represents the dot product of two vectors equally a multiplication of the ii vectors, and the cosine of the angle formed between them. Dot product formula of the given vectors can exist expressed as follows. Here a and b are the two vectors, |a| and |b| are their respective magnitudes, and θ is the angle between the two vectors a.b = |a||b| cos⁡θ.

Does Dot Product Formula Involve Multiplication?

Multiplication of two vectors is non the same as scalar multiplication. In that location are 2 types of multiplication involving two vectors. The scalar product is 'dot product' and the vector product is 'cross product'. The dot product formula represents the dot product of two vectors as a multiplication of the 2 vectors, and the cosine of the angle formed between them.

What is the Purpose of Dot Product Formula?

The purpose of the dot production is to tell us the amount of force vector is applied in the direction of the move vector. The dot product also lets the states mensurate the angle that is formed past a pair of vectors and the relative position of a vector against the coordinate axes.

What Happens When a Dot Production after Using Dot Product Formula Is 0?

The dot product formula represents the dot product of two vectors every bit a multiplication of the 2 vectors, and the cosine of the angle formed between them. If the dot product is 0, then we can conclude that either the length of one or both vectors is 0, or the angle between them is ninety degrees.

Where Do Nosotros Apply Dot Product?

The concept of the dot product is used prominently in physics and engineering. For 2 quantities placed at an angle to each other, the dot product gives the result of these two vectors. Let us have an case of force applied on a body F, and the displacement of the body is d. If the angle betwixt the force vector F and the deportation vector d is θ, then the work done is the product of force and displacement. Due west = F.d.Cosθ.

How To Calculate the Dot Product?

The dot product can be calculated in 3 unproblematic steps. First detect the magnitude of the two vectors a and b, ie |a| and |b|. Secondly, find the cosecant of the bending θ between the two vectors. Finally take a product of the magnitude of the ii vectors and the and cosecant of the angle betwixt the two vectors, to obtain the dot product of the 2 vectors. (a.b = |a|.|b|.Cosθ. Also cheque to dot product computer, to easily detect the vector dot production.

Why is the Dot Product Chosen Scalar Production?

The dot product is a scalar considering all the individual constituents of the answer are scalar values. In a.b = |a|.|b|.Cosθ, |a|, |b|, and Cosθ are all scalar values. Hence the dot product is also chosen a scalar product.

Why Do We Use Cos in Dot Product?

For finding the dot product we demand to have the ii vectors a, b in the same management. Since the vectors, a and b are at an angle to each other, the value acosθ is the component of vector a in the direction of vector b. Hence we can observe cosθ in the dot product of 2 vectors.

Why is the Dot Product of Orthogonal Vectors Equal to 0?

The 2 orthogonal vectors are perpendicular to each other and the bending between the ii vectors is equal to 90°. Since Cos90° = 0, the dot product of ii orthogonal vectors is equal to 0. a.b = |a|.|b|.cos90° = |a|.|b|.0 = 0.

Why is the Dot Product Commutative?

The dot product of two vectors is equal to the product of the magnitude of the two vectors and the cosecant of the angle between the two vectors. And all the individual components of magnitude and bending are scalar quantities. Hence a.b = b.a, and the dot product of vectors follows the commutative property.

Tin can a Dot Production be equal to nix?

The dot production of ii vectors can be zippo if either of the ii vectors is zip or if the two vectors are perpendicular to each other. For two non-zero vectors, the dot production is zero if the angle betwixt the two vectors is 90º, because Cos90º = 0.

Source: https://www.cuemath.com/algebra/dot-product/

Posted by: jonesbuzzlieve.blogspot.com