Scalar Products

The scalar product, sometimes known as the dot product, is an algebraic operation that yields a single integer from two equal-length sequences of numbers. A scalar product results in a scalar quantity or a real number. It is calculated similarly to an algebraic operation. 

The most common application of scalar products in physics and astronomy is multiplying two separate vectors. A vector is an entity with both magnitude and direction, while a scalar only possesses magnitude.

Keep reading these study material notes on scalar products to learn everything there is to know about it.

Scalar Product of Two Vectors

The scalar product of two vectors is the product of the magnitudes of the two vectors and the cosine of their angle.

In other words, the scalar product is the sum of the first vector’s magnitude and the projection of the first vector onto the second vector. For two vectors a and b, the scalar product formula is as follows:

   a.b = |a| |b| cosθ

cosθ = a.b / |a||b|

      θ = cos−1(a.b / |a||b|)

When a and b are two non-zero vectors and θ is the angle between them, then their scalar product (or dot product) is denoted by a.b and is defined as the scalar |a||b|cosθ, where |a| and |b|are moduli of a and b respectively and 0≤θ≤π. 

When we express the same vectors in the form of unit vectors I, j and k, respectively, we define the scalar product as follows:

a = a1i+a2j+a3k       

b  = b1i+b2j+b3k

Applications of Scalar Product

In vector theory, the scalar product has a variety of applications which includes the following:

• Projection of a Vector: 

We can use the scalar product to calculate a vector’s projection onto another vector. 

For determining the projection of a vector “a” onto vector “b”, the formula is a.b / |b|. Conversely, the projection of vector “b” onto vector “a” is given by a.b / |a|.

• Scalar Triple Product: 

The scalar product is useful to determine the scalar triple product of three vectors. Following is the formula for the scalar triple product: 

1. (b × c) = b. (c × a) = c. (a × b)

• Angle Between Two Vectors: 

The formula below helps calculate the angle between two vectors using the scalar product.

cos θ= (a.b)/(|a| |b|)

Scalar Product Properties

Now that you’ve grasped the notion of scalar product, let us look at some key features of the scalar product of vectors a and b that can aid you in solving problems:

• Commutative Law: 

This principle or law asserts that the multiplication of two real numbers remains unchanged when you reorder them. We may say the same for vectors. Even after rearranging vectors when extracting their product, the outcome of a scalar product stays unaffected. Therefore: 

a.b = b.a

• Distributive Law: 

Suppose we multiply a number by a sum of numbers. Then, the result will be the same as if we multiply the number by each number separately and add the products together. This function is the distributive law. 

We may calculate the scalar product of vectors using this distributive property. Therefore: 

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

• Scalar Multiplication: 

The scalar product fulfils the scalar multiplication in the following way:

→    →      →  →

(r1a) . (r2b) = r1r2 (a.b)

• If the scalar product of two vectors is zero, they are orthogonal. Therefore, if a.b = 0, vectors a and b are orthogonal. It also means that the vectors a and b are perpendicular to each other. Therefore, when you consider three orthogonal unit vectors: i, j, and 

1. j = j.k = k.i = 1.1Cos 90° = 0  ∴ 

• When cos θ = 1, i.e. θ = 0°, the scalar product of two vectors will be maximum, i.e., when the vectors are parallel. 

   → →

( a.b)max = ab

• When cos θ = -1, i.e. θ = 180°, the scalar product of the two vectors will be minimum. Therefore, they are not parallel or antiparallel.

 → →

( a.b)min= -ab 

Conclusion

Let us recapitulate. We may class most of the quantities we are familiar with as either a scalar or a vector quantity. The distinction between scalar and vector values is significant. Scalar quantities have simply a magnitude and no direction to them. 

When trying to get the scalar product of two vectors, take one vector in the direction of the other and multiply it with the magnitude of the first one. We cannot compute the scalar product for vector quantity if the direction and magnitude are absent.