Formula For Vectors Dot Product

Vector is a term used to signify a group of quantities or lines that have a numerical magnitude along with a fixed direction associated with it. Vectors hold great importance in physics as well as in mathematics. Vectors are primarily involved in the Conic section as problem-solving methods that have to deal with figures. Vector multiplication is a crucial process when calculating the resultant output of two vectors. There are two kinds of multiplication that a vector can go through, namely a dot product and a cross product. In this article, we will explain the dot product of two vectors and sort simple queries such as trigonometric functions like cos used in the dot product of two vectors.

What Is The Dot Product Of Two Vectors?

Dot Product is one of the ways to multiply two distinct or equal vectors. The primary purpose of using dot product in multiplying any two vectors is to calculate the magnitude of the resultant vector. In Physics, a dot product is also called a scalar product between two vector quantities. 

The theoretical definition of dot product can be expressed as the multiplication of the respective vectors’ magnitudes and the cosine angle made by the two vectors in between them. 

Suppose there are two vectors, namely, vector a and vector b, which are non-zero vectors. Then, the dot product is denoted between a and b is a . b. Here, |a| denotes the magnitude of the absolute value of vector a, and |b| denotes the magnitude of the absolute value of vector b. θ is the angle generated between a and b vectors in the anti-clockwise direction. 

  a . b = |a||b|*cosθ 

Note: The scalar product or the dot product of two vectors, if represented in a three-dimensional graph, will lie in the same plane, just like the two multiplied vectors.

Geometrical Understanding Of Dot Products

The dot products of two vectors can be understood geometrically more easily than theoretically. After all, the scalar product or dot product has been of supreme importance in the physics calculation of two vector quantities. 

If you forget the dot product formula, at what time is the cos used in the dot product. Here are a few sentences that can help you remember them. Take the component of one vector, let’s say a, in the direction of vector b and then multiply it by the magnitude of vector b.

Magnitude Of A Vector

The magnitude of a vector is the numerical equivalent to the result obtained by applying the absolute value function on the vector. Since we know that a vector comprises one magnitude and one direction, we only need to worry about the magnitude here because it is a scalar product. 

Let’s assume two vectors a = a1i + a2j + a3k and b = b1i + b2j + b3k.

a1, a2 and a3 are three magnitude coefficients of vector a.

Similarly, b1, b2, and b3 are three magnitude coefficients of vector b.

I, j, and k are the direction along the x-axis, y-axis, and z-axis.

The magnitude of a vector is always a positive quantity. 

The magnitude of vector a is calculated by,

|a| =  √(a1)2 + (a2)2 + (a3)2

The magnitude of vector b is calculated by,

|b| = √(b1)2 + (b2)2 + (b3)2

Projection And Angle Of Two Vectors

To understand the projection of a vector in a simple manner, we will take two vectors a and b.

So, If we place vector a on vector b from the point of their intersection, then the length of vector b occupied by vector a is the projection of a vector a on vector b. It is like the shadow of vector a falling onto vector b. 

• The projection of a vector a on the vector b is calculated by, 

(a.b)/|b|

• The projection of a vector b on the vector a is calculated by,

(a.b)/|a|

• The angle between any two vectors formed by their intersection at one point is calculated by 

Cosθ = (a.b)/|a|*|b|

Magnitude Of The Dot Product Of Two Vectors 

Suppose there are two vectors, P and Q.

P = a1i + a2j + a3k , and Q = b1i + b2j + b3k

Then, P.Q will define the scalar product.

P.Q = (a1i + a2j + a3k).(b1i + b2j + b3k)

P.Q = {(a1b1)*(i.i) + (a1b2)*(i.j) + (a1b3)*(i.k)} + {(a2b1)*(j.i) + (a2b2)*(j.j) + (a2b3)*(j.k)} + {(a3b1)*(k.i) + (a3b2)*(k.j) + (a3b3)*(k.k)} 

P.Q = a1b1 + a2b2 + a3b3

Dot Product Operations On Vectors

1. Commutative: – If we have the dot product of vector a and vector b with this order (a.b), then the dot product of vector b and vector a in the order (b.a) will be the same.

a.b = b.a = |a||b|cosθ = |b||a|cosθ

2. Distributive: – Distributive property is also valid in the dot product of two vectors or any number of vectors. To formulate the dot product formula for the distributive property, we take three vectors a, b, and c.

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

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

a.(b-c) = (a.b – a.c)

(a-b).c = (a.c – b.c)

Conclusion

The dot product of two vectors has multiple applications in the study of mathematics as well as physics. They are used to calculate many physics formulas such as force, work, etc. They are also used in conic sections topics such as the straight lines or vector 3d because it requires projection problems. Hence, you learned all the important formulas of the dot products between two vectors.