|Table of contents|
|Dot Product Definition|
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The dot product of two different vectors that are non-zero is denoted by a.b and is given by:
a.b = ab cos θ
wherein θ is the angle formed between a and b, and,
0 ≤ θ ≤ π
If a = 0 or b = 0, θ will not be defined, and in this case,
You can define the dot product of two vectors in two different methods: geometrically and algebraically.
The geometric meaning of dot product says that the dot product between two given vectors a and b is denoted by:
a.b = |a||b| cos θ
Here, |a| and |b| are called the magnitudes of vectors a and b and θ is the angle between the vectors a and b.
If the two vectors are orthogonal, that is, the angle between them is 90, then a.b = 0 since cos 90 = 0.
If the two vectors are parallel to each other, then a.b =|a||b| since cos 0 = 1.
The dot product algebra says that the dot product of the given two products – a = (a1, a2, a3) and b= (b1, b2, b3) is given by:
a.b= (a1b1 + a2b2 + a3b3)
Properties of Dot Product of Two Vectors
Given below are the properties of vectors:
Dot Product of Vector-Valued Functions
The dot product of vector-valued functions, that are r(t) and u(t), each gives you a vector at each particular time t, and hence, the function r(t)⋅u(t) is said to be a scalar function.
Example 1: Find the dot product of a= (1, 2, 3) and b= (4, −5, 6). What kind of angle the vectors would form?
Using the formula of the dot products,
a.b = (a1b1 + a2b2 + a3b3)
You can calculate the dot product to be
= 1(4) + 2(−5) + 3(6)
= 4 − 10 + 18
Since a.b is a positive number, you can infer that the vectors would form an acute angle.
Example 2: Two vectors A and B are given by:
A = 2i − 3j + 7k and B= −4i + 2j −4k
Find the dot product of the given two vectors.
A.B = (2i − 3j +7k) . (−4i + 2j − 4k)
= 2 (−4) + (−3)2 + 7 (−4)
= −8 − 6 − 28
Key Points to Remember
Vector is a quantity that has both magnitude as well as direction. Few mathematical operations can be applied to vectors such as addition and multiplication. The multiplication of vectors can be done in two ways, i.e. dot product and cross product. The dot product of two vectors is the sum of the products of their corresponding components. It is the product of their magnitudes multiplied by the cosine of the angle between them. A vector's dot product with itself is the square of its magnitude.