Dot Product of Opposite Vectors

Vectors can be defined as some quantity that distinguishes both magnitude and the direction of the body in a particular direction. The word ‘Vector’ has been derived from the Latin word ‘Vectus’, which implies to carry. Most commonly known vector quantities are force, acceleration, momentum and weight, which provides both magnitude and the direction of an object. It should also be noted that the distance between the introductory and the terminal point, magnitude, is indicated by two parallel lines | |. As we do multiplication in basic mathematics, one can also form a product of vectors which is also known as the dot product between two vectors. 

Define Dot product

A dot product is defined as the product of two vectors. The resultant we get out of this multiplication is a scalar quantity, thus the dot product is also referred to as a scalar product. Additionally, we can comprehend this dot product property of vectors as the sum of the product of the interconnected units in two vectors. 

A.B = a0b0 + a1 b1 + a2 b2 + a3 b3

We get the dot product of vectors A and B by multiplying the magnitude values of the two vectors with the cosecant of the angle that is formed with the adjoining of the two vectors. Unlike magnitude, the dot product can either be a positive real-valued number or a negative one. 

A.B = |a||b| cos θ

In this formula, |a| is the magnitude of vector A and |b| is the magnitude of vector B. Cos θ defines the angle that is formed between the two vectors. We can find the value of Cos θ with the help of the formula: 

  Cos θ = (a0 b0 + a1 b1 + a2 b2 + a3 b3)/√(a0²+  a1²+ a2²). √(b0² + b1² + b2²)

Geometric components of dot product of two vectors

Geometrically, the dot product between two vectors is a combination of distinct values of the magnitude of the two vectors and the vector projection of one vector over another. 

The magnitude of a Vector 

The magnitude of a vector can be defined as the square root of the aggregate of the distinctive units or variables of a vector quantity. Notably, It should be remembered that the value of the magnitude is always a positive quantity. We depict magnitude by the formula √(a02 + a1² + a2²).

Vector projection 

The projection of a vector pertains to the promontory of the shadow of one vector over the other vector. We determine the vector projection by dividing the product of the magnitude of the vectors and with the magnitude and direction of the first vector, that is vector A. 

Dot product properties

There are various properties of a dot product of two vectors. One can simply comprehend and understand this with basic essentials of properties of addition, subtraction and multiplication. Some of the dot product properties are: 

1) Commutative property of Vectors

You might comprehend this better from what you have already study in previous years of classes is the properties of multiplication where even if we change positions for numbers, the resultant value or the product would still be the same. 

Here, C.D is the same as D.C in vectors or we can say that the dot product of vector C with D and the dot product of vector D with C always gives us the same result. So, we can depict this commutative property of the dot product between two vectors as: 

 Vector C. Vector D = Vector D. Vector C

2) Distributive property of vectors

As with the case of addition and multiplication in the above case, the same is with the dot product of vectors C and D. Here, if we deduce a dot product of vector A with the vector formed by adding Vector B and C, then that would be equal to the dot product of the sum of A and B with C. 

   Vector [A.(B+C)] = Vectors [(A+B).C]

3) Other properties 

•The first one is the point that if a vector is multiplied with itself then the resultant vector quantity would be the square of the given vectors.

•Next one is to comprehend about the length of a given vector, let it be C, which is known by the square root of the product of the vector with itself. 

•Notably, if a unit vector is multiplied with itself then the resultant dot product between two vectors would be 1 as the value of θ, in this case, would be 0. 

•However, if two vectors are perpendicular to each other, then the resultant dot product would be 0. 

Exemplary questions for the Dot product between two vectors 

Question 1: Suppose there are two vectors that intersect each other, perpendicularly. One vector quantity is upright on the other vector quantity. Find the dot product of these two vectors.

Answer: In this case, if two distinct vectors are perpendicular to each other, then the resultant dot product between two vectors would be 0.

Since the product of these two vectors A and B is a positive value, thus the angle that would be formed by these vectors would be an acute angle. 

Question 2: Suppose there are two vectors C and D. C is defined as 2i – j + 3k and D is defined as 4i + 2j – k. Find the dot product of vectors C and D. 

Answer: We can comprehend this as the sum of the product of the corresponding units in two vectors. 

C.D =  (2i – j + 3k).( 4i + 2j – k)

C.D = 2×4 + (-1)×2 +3×(-1)

C.D = 8 – 2 – 3

C.D = 3

Question 3: Find the magnitude of the vector C if it is defined as 2i + 3j + 2k. 

Answer: The magnitude of a vector is defined as the square root of the sum of the individual units of a vector. It is given by the formula √(c0² + c1² + c2²).

|C| = √[(2)² + (3)² + (2)²]

|C| = √[4 + 9 + 4]

|C| = √17

Conclusion 

Thus, the dot product between two vectors is defined as the multiplication or the product of the given vector. One can easily comprehend the dot product of vectors if both these values are known. It is important that one understands and comprehends the properties of dot products with basic essentials of properties of addition, subtraction and multiplication.