## Dot Product of Two Vectors

Scalar Dot Product Of Two Vectors: Dot Product of Two Vectors – In order to understand the Dot product of two vectors, we need to first understand what a projection is. A projection of one vector on the other is often referred to as the orthogonal projection of one vector on the other.

The dot product of two vectors is a scalar quantity that is calculated by multiplying the magnitude of one vector by the magnitude of the other vector and then by the cosine of the angle between them. The dot product is a measure of the similarity of two vectors and is use in physics to calculate the force between two objects.

## Dot Product of Two Vectors – Explained

The dot product of two vectors is a scalar value that compute by multiplying the components of each vector and then summing the products. The result is a number that represents the magnitude of the vector product and the direction is given by the angle between the vectors.

## Dot Product Definition

A dot product is a mathematical operation that takes two vectors and multiplies them together to produce a single scalar value. The dot product denote by the symbol . The two vectors place side by side, with the first vector on the left and the second vector on the right. The dot product calculate by multiplying each component of the first vector by the corresponding component of the second vector and then adding all of the resulting products together.

## Dot Product Formula

The dot product is a mathematical formula that calculates the product of two vectors. The formula is:

Vector a dot Vector b = |a| |b| cos(θ)

In this formula, a and b are the vectors, |a| is the magnitude of a, and |b| is the magnitude of b. θ is the angle between a and b.

## Dot Product Geometry Definition

In mathematics, the dot product is an operation that takes two vectors in a three-dimensional space and returns a single number. The dot product define as the sum of the products of the corresponding components of the vectors.

## Dot Product Algebra Definition

The Dot Product Algebra Definition is a mathematical operation that calculates the product of two vectors. It represent by the symbol “.” and is usually calculate using the Cartesian coordinate system.

## Properties of Dot Product of Two Vectors

In mathematics, the dot product is a binary operation that takes two vectors, u and v, and produces a scalar quantity, often denoted by 〈u, v〉, that is the product of the magnitude of u and the magnitude of v, and the cosine of the angle between them.

The dot product is distributive, meaning that 〈u, (v + w)〉 = 〈u, v〉 +

## Dot Product of Vector-Valued Functions

Let be a vector-valued function on an open subset of . The dot product of and is define as

where is the norm of .

We will show that is a linear function. That is, for any two vectors and and any real numbers and ,

The proof is by induction on . The base case is easy, since

The inductive step is also easy, since

for all vectors and all real numbers .

## Solved Examples

Question:

What is the difference between an open and closed economy?

An open economy is one in which the government allows foreigners to invest in the country and purchase its assets. A closed economy is one in which the government does not allow foreigners to invest in the country and purchase its assets.

## NCERT Study Material

NCERT is a government organisation which provides study material to students of classes I to XII. The NCERT study material is very helpful for students as it helps them in understanding the concepts taught in school in a better way.

The NCERT study material is available in both Hindi and English. It is very well-organise and easy to understand. The NCERT study material includes text books, work books, teachers’ manuals and activity books.

The text books are written