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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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.
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.
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 \[\widehat{X}\] (\[\widehat{Y}\]+ \[\widehat{Z}\]) = \[\widehat{X}\] . \[\widehat{Y}\] + \[\widehat{X}\] . \[\widehat{Z}\]
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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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 * \[\widehat{Y}\])
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 = √(22 + 32 = √4 + 9 = √13
YB= √32 + (-4)2 = √9 + 16 = √25 = 5
The cross product A X B = AB Sin θ = 5 * √13 * Sin 60 = 5*√13*√3/2
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]
Y=
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.
[Xx Xy Xz] = XxYx + XyYy + XzYz = X. Y
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. |