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There are two types of quantities that are associated with bodies in motion, either scalar quantities or vector quantities. The magnitude of a scalar quantity does not have any relationship to its direction, whereas the magnitude and the direction are the same for a vector quantity. The vector can be used to perform multiple mathematical operations including addition and multiplication. It’s simple to add vectors but to multiply them, we need to do both a dot product and a cross product. In vector dot products, the result is always a scalar quantity, so the scalar product is also called the dot product. In this article, we will look more closely at the dot products of two vectors and use solved examples to explain how they work.
What is a Dot Product of Vectors?
As a result of multiplying two vectors, we can define the dot product as the magnitude of each vector multiplied by the cosine of the angle between them. When two vectors are multiplied, their result or the dot product is in the same plane as the two vectors. A scalar number is obtained by performing a mathematical operation on the vector components. When two vectors are identical in the number of dimensions, the dot product is applied. In mathematics, the dot product of a vector with itself is equal to twice its magnitude. A dot represents the dot product of two vectors.
Definition of Dot Products?
Two vectors a and b with magnitudes |a| and |b| are produced by adding their dot products or scalar products, where 𝜃 represents the angle between the vectors. Vectors are bound by their dot products and expressed as:
a.b = |a| |b| cos 𝜃
where,
|a| represents the magnitude of vector a.
|b| represents the magnitude of vector b.
cos 𝜃 represents the cosine of the angle between two vectors (0 <= 𝜃 <= ℼ).
a.b will not be defined in the case if a and b are both equal to 0.
a.b indicates the dot product of two vectors.
Geometrical Representation of Dot Product
We multiply the magnitude of one vector by its component so as to find the dot product of two vectors. Our vectors are multiplied according to the direction they are moving. By finding their magnitude over each other and the angles between them, one can determine the vector dot product.
Angle Between Two Vectors Using Dot Product
An angle between two vectors can be determined by calculating the cosine of their angle. By taking the sum of the individual components of two vectors and dividing it by their magnitude angle between two vectors.
Properties of Dot Product of Two Vectors
The properties of vectors are as follows:
Commutative Property:
a .b = b.a
a.b =|a| b|cos θ
b.a =|b||a|cos θ
Distributive Property:
a.(b + c) = a.b + a.c
Bilinear Property:
a.(rb + c) = r.(a.b) + (a.c)
Scalar Multiplication Property:
(xa) . (yb) = xy (a.b)
Non-Associative Property:
This is because the dot product of a vector and a scalar is not permitted.
Orthogonal Property:
If a.b = 0, then two vectors are orthogonal
Solved Examples of Dot Product
Example 1: a=1, 2, 3 and b=4, *5, 6; find their dot products. Which angle would result from the vectors?
Solution:
Based on the formula of the dot products,
The following expression is equivalent to (a1b1 + a2b2 + a3b3)
The dot product can be calculated as follows
= 1(4) + 2(−5) + 3(6)
= 4 − 10 + 18
= 12
In the case where a.b is a positive number, the angle formed by the vectors would be acute.
Example 2:
A and B are the two vectors that have the following properties: The formula for A equals 2i * 3j + 7k, and the formula for B equals *4i + 2j * 4k. Using the given vectors, find the dot product.
Solution:
A.B = (2i − 3j +7k) . (−4i + 2j − 4k)
= 2 (−4) + (−3)2 + 7 (−4)
= −8 − 6 − 28
= −42
Example 3: Suppose there are two vectors [6, 2, -1] and [5, -8, 2]. Calculate the dot product of the two vectors.
Solution:
Let [6, 2, -1] and [5, -8, 2] respectively be the vectors a and b.
a.b = (6)(5) + (2)(-8) + (-1)(2)
a.b = 30 – 16 – 2
a.b = 12
Example 4: Let a and b be two vectors equal to 4 and 2 respectively, and let θ be 60 degrees. Find the dot product of the two.
Solution:
a.b = |a||b|cos θ
a.b = 4.2 cos 60°
a.b = 4.2 × (1/2)
a.b = 4
Conclusion
A dot product, or inner product, is created when two vectors have specific components. To calculate the product, multiply the magnitudes by the sine of the angle between them. When a vector is added to itself, the dot product will be the square of its magnitude. Dot products do not require the vectors to be in commutative order; therefore, the vectors need not be arranged commutatively. Adding a constant to a vector has the same dot product as adding another vector to it. Dot products are zero when vectors are multiplied by zero vectors. A pair of nonzero vectors can only be orthogonal or perpendicular if their dot product equals 0. This article presented a mathematical and geometrical understanding of dot products of two vectors and also demonstrated how they are solved in some examples.
