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Scalar and Vector Products

Last updated date: 19th May 2024
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Introduction to Scalar and Vector Products

The underlying concepts of Physics have a mathematical base. All measurable quantities are physical quantities. The motion of objects can be described by two mathematical quantities: a scalar and a vector.

  • A scalar quantity is described completely by magnitude or numbers alone. Examples of scalar quantities are length, mass, distance, energy, volume, etc.

  • A vector quantity needs a magnitude as well as a direction to describe it completely. Examples of vector quantities are displacement, velocity, weight, dipole moment, etc.


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Vector Quantities Can Be Multiplied in Two Ways

  • Scalar or dot product

  • Vector or cross product


In this article, we will discuss scalar and vector products and solve a few examples where we will find the scalar and vector product of two vectors.


Define Scalar Product of Two Vectors

The scalar product of two vectors gives you a number or a scalar. Scalar products are useful in defining energy and work relations. One example of a scalar product is the work done by a Force (which is a vector) in displacing (a vector) an object is given by the scalar product of Force and Displacement vectors. The scalar product is denoted by a dot(.) and the formula of scalar product is given below:

\[\widehat{X}\] . \[\widehat{Y}\] = XY Cos ፀ, where ፀ is the angle between the vectors.

The scalar product is also called the dot product because of the dot notation used in it. 


Properties of Scalar Product of Two Vectors

  • The direction of the angle ፀ has no significance in the dot product of two vectors. The angle ፀ can be measured from either of the vectors to the other since Cos ፀ = Cos (-ፀ) = Cos (2ℼ - ፀ)

  • If ፀ is more than 90 degrees and less than or equal to 180 degrees then the dot product is a negative value i.e. 900 < ፀ <= 1800

  • If ፀ is more than 0 degrees and less than or equal to 90 degrees then the dot product is a positive value. i.e. 00 < ፀ <= 900

  • The dot product of two vectors that are parallel to each other is given by  \[\widehat{X}\] . \[\widehat{Y}\]= XY Cos 0 = XY.

  • The scalar product of two anti-parallel vectors is given by \[\widehat{X}\] . \[\widehat{Y}\] = XY Cos 180 = -XY.

  • The scalar product of a vector multiplied by itself is the square of its magnitude. \[\widehat{X}\] . \[\widehat{X}\] = XX Cos 0 = X2

  • The scalar product of two orthogonal vectors is 0 i.e. \[\widehat{X}\] . \[\widehat{Y}\]= XY Cos 90 = 0


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  • The dot product is commutative i.e. the order of the two vectors in the product does not matter. So, \[\widehat{X}\] . \[\widehat{Y}\] = \[\widehat{Y}\]. \[\widehat{X}\]

  • The dot product is distributive which means \[\widehat{X}\]  (\[\widehat{Y}\]+ \[\widehat{Z}\]) = \[\widehat{X}\] . \[\widehat{Y}\] + \[\widehat{X}\] . \[\widehat{Z}\]


Define Vector Product of Two Vectors

When we take the vector product of two vectors, we get a vector. The Vector product is also termed as the cross product as the sign for the vector product is a cross(X)

\[\widehat{X}\] X \[\widehat{Y}\]

The direction of the vector product of two vectors is perpendicular to both the vectors. This means that the cross product of two vectors \[\widehat{X}\] and \[\widehat{Y}\]  lies in a plane that is perpendicular to the plane which contains Xand Y. The formula to give the magnitude of the vector product is:

| \[\widehat{X}\] x \[\widehat{Y}\] | = XY *Sin θ. Here the angle θ between the vectors is measured from the first vector in the formula (here vector X) to the second vector (vector Y) in the formula.


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Properties of Cross Product of two Vectors

  • The angle between the vectors,  θ, lies between 0 and 180 degrees.

  • The vector product of vectors which are parallel to each other (where  θ = 0) or antiparallel to each other (where  θ = 180) is 0 since Sin 0 = Sin 180 = 0

  • The resultant vector of the cross product of the two vectors could lie either on the upward or downward plane. 

  • The vectors \[\widehat{X}\] X \[\widehat{Y}\]and \[\widehat{Y}\] X \[\widehat{X}\] are antiparallel to each other hence vector product is not commutative.

  • If the order of multiplication is changed, the resultant vector changes in sign i.e \[\widehat{X}\] X \[\widehat{Y}\]= - \[\widehat{Y}\] X \[\widehat{X}\].

  • The common mnemonic used to determine the direction of the cross product of vectors is the corkscrew right-hand rule. The direction of the vector is given by turning the corkscrew handle from the first to the second vector.


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  • The length of the vector product of two vectors equals the area of the parallelogram determined by the vectors.


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  • The cross product of two vectors is distributive i.e. \[\widehat{X}\]  X  (\[\widehat{Y}\]+ \[\widehat{Z}\] ) =  \[\widehat{X}\]  X  \[\widehat{Y}\] +  \[\widehat{X}\] X  \[\widehat{Z}\].

  • The multiplication by a scalar satisfies (k *  \[\widehat{X}\]) X  \[\widehat{Y}\] = k * ( \[\widehat{X}\] X  \[\widehat{Y}\]) =  \[\widehat{X}\] X (k * \[\widehat{Y}\])


Solved Examples of Scalar and Vector Product of Two Vectors

Let us find the scalar and vector product of two vectors through a couple of examples:

  • For which real number r the vectors X and Y in the equation given below are perpendicular to each other: X = (-2, -r) and Y = (-8, r)

Solution - If two vectors are perpendicular to each other then their scalar product is 0. So we get:

(-2)(-8) + (-r)(r) = 0 i.e. r2 = 16, hence r = 4 or -4.

  • What is the cross product of two vectors A = 2i + 3j and B = 3i - 4j which have an angle of 60 degrees between them.

Solution - we first find the magnitude of the two vectors:

A = \[\sqrt{2^2 + 3^2}\] = \[\sqrt{4 + 9}\] = \[\sqrt{13}\]

YB= \[\sqrt{3^2 + (-4^2)}\] =  \[\sqrt{9 + 16}\] = \[\sqrt{25}\] = 5 

The cross product A X B = AB Sin θ = 5 * \[\sqrt{13}\] * Sin 60 = 5*\[\sqrt{13}\]*\[\sqrt \frac {3}{2}\]

FAQs on Scalar and Vector Products

1. How do you represent the scalar product of two vectors as a matrix?

It is sometimes useful and convenient to represent vectors as either row or column matrices rather than as unit vectors. If we represent the x, y, and z coordinates of a vector as a column matrix then we would get row matrices by transposing them. So we could write:

XT = [Xx Xy Xz

Now if we take the matrix product of the above two vectors, it would give us a single number which is the scalar product of the two vectors.

2. Differentiate between a scalar and a vector quantity

Scalar Quantity

Vector Quantity

These only have a magnitude

To express them we need both the magnitude and the direction of these quantities.

This is always a positive number

This can be positive or negative

We can use simple mathematical methods to add, multiply, divide, and subtract scalar quantities.

We need to employ complex algebra to add, multiply, subtract, or divide vector quantities.

3. What is the relationship between scalar and vector products?

A vector can be multiplied by another vector, but not split. There are two kinds of vector products that are generally used in physics and engineering. A scalar multiplication of two vectors is one type of multiplication. As the name implies, the scalar product of two vectors yields a number (a scalar). The relationship between energy and work are defined using scalar products. A scalar product of the force vector and the displacement vector, for example, defines the work that a force (a vector) does on an object while generating its displacement (a vector). A vector multiplication of vectors is a very different type of multiplication. Taking the vector product of two vectors yields a vector, as the name implies. Other derived vector quantities are defined using vector products. A vector variable termed torque, for example, is defined as the vector product of an applied force (a vector) and its distance from the pivot to the force in defining rotations (a vector). Because a scalar product is a scalar quantity and a vector product is a vector quantity, it is crucial to distinguish between the two types of vector multiplications. 

4. How are vectors utilized in physics and engineering?

In the physical sciences, vectors are essential. They can be used to represent any quantity that has magnitude, direction, and follows the vector addition rules. One example is velocity, which has the magnitude of speed. For example, the vector (0, 5) (in two dimensions with the positive y-axis as 'up') could represent a velocity of 5 metres per second upward. Force is another quantity that may be represented by a vector because it has a magnitude and direction and satisfies the vector addition principles. Many other physical quantities are described using vectors, including linear displacement, displacement, linear acceleration, angular acceleration, linear momentum, and angular momentum. Other physical vectors, such as the electric and magnetic fields, are represented as a vector field, which is a system of vectors at each point of a physical space. 

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