A dot product is an algebraic sum of the products of two sequences of numbers, whereas a geometric dot product is the sum of the Euclidean magnitudes of two vectors plus the cosine of the angle between them. The definitions are equivalent when using Cartesian coordinates, however, vector spaces are usually used when defining Euclidean spaces in modern geometry. Dot products are used in this situation to define lengths (the square root of the dot product of each vector’s length) and angles (the cosine of an angle formed by two vectors, x, and y are vectors whose dot product is their product).

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## Properties of Dot Product

**Distributive Property**

a.(b + c) = a.b + a.c

**Bilinear Property**

a.(rb + c) = r.(a.b) + (a.c)

**Commutative property**

a .b = b.a

a.b =|a| b|cos θ

a.b =|b||a|cos θ

**Non-Associative Property**

Since dot product between a scalar and a vector is not allowed.

**Orthogonal Property**

Only if a and b are orthogonal are two vectors considered orthogonal.

**Scalar Multiplication Property**

(xa) . (yb) = xy (a.b)

## Formula to Calculate Dot Product

The dot product of two vectors is given in the formula →a. →b=|a||b|cos(θ) a → . b → = | a | | b | cos The dot product of two vectors is a scalar and lies in the plane of the two vectors.

## Dot Product of Two Vectors

Dot product is the sum of the products of the opposite entries of the two sequences of vectors is equal to the product of magnitudes of two vectors and cosine of the angle between them, the resultant of the dot product of two vectors either can be positive or can be negative in nature, The dot product of two vectors lies in the same plane.

**Example 1:** When you divide a by b, find the dot product of a and b equals (1, 2, 3). What kind of angle does that create?

**Solution:**

Using the formula of the dot products,

a.b = (a1b1 + a2b2 + a3b3)

You can calculate the dot product

= 1(4) + 2(−5) + 3(6)

= 4 − 10 + 18

= 12

**Example 2:** Calculate the dot product of c=(−4,−9)

and d=(−1,2). The vectors form which angle, an acute angle, right angle, or obtuse angle?

**Solution:** The dot product of two-dimensional vectors can be computed by applying the component formula.

a⋅b=a1b1+a2b2,

we calculate the dot product to be

c⋅d=−4(−1)−9(2)=4−18=−14.

The geometric definition indicates that c*d is a negative number, so the angle formed by the vectors is obtuse.

**Example 3:** If a=(6,−1,3), then what value of c is the vector b=(4,c,−2) perpendicular to a?

**Solution:** It is essential that their dot product is equal to zero for a and b to be perpendicular. Since

a⋅b=6(4)−1(c)+3(−2)=24−c−6=18−c,

the number c must satisfy 18−c=0

18−c=0, or c=18.

You can double-check that the vector b=(4,18,−2)

the line is in fact perpendicular to a by verifying that

a⋅b=(6,−1,3)⋅(4,18,−2)=0.

## Key Points:

- The output of two vectors being cross-product is an orthogonal vector
- When two vectors are cross-products, the right-hand thumb rule determines the direction of the cross product, while the magnitude comes from the area of the parallelogram created by the original pair of vectors.
- In mathematics, zero vectors are the cross-product of two linear vectors.

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