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. What is the main difference between scalar and vector quantities?
The main difference lies in their properties. A scalar quantity is defined by its magnitude (a numerical value) alone, for example, a speed of 50 km/h. In contrast, a vector quantity requires both magnitude and direction for its complete description, for instance, a velocity of 50 km/h heading East. Other examples of scalars include mass and temperature, while force and displacement are vectors.
2. How is the scalar product (or dot product) of two vectors defined and calculated?
The scalar product of two vectors, say A and B, results in a scalar quantity. It is calculated by multiplying the magnitudes of the two vectors with the cosine of the angle (θ) between them. The formula is expressed as A · B = |A| |B| cos(θ). This product essentially measures how much of one vector is projected along the direction of the other.
3. What is the vector product (or cross product) of two vectors, and what does it represent?
The vector product of two vectors, A and B, results in a new vector that is perpendicular to the plane containing both A and B. Its magnitude is calculated using the formula |A × B| = |A| |B| sin(θ), where θ is the angle between them. The direction of this new vector is determined by the right-hand thumb rule, and it is crucial for defining rotational quantities like torque.
4. Why is the scalar product often called the 'dot product' and the vector product the 'cross product'?
These names originate from the mathematical symbols used to denote them. The scalar product is written with a dot symbol between the vectors (e.g., A · B), hence the name 'dot product'. Similarly, the vector product uses a cross symbol (e.g., A × B), which is why it is commonly referred to as the 'cross product'. These symbols are a standard convention to visually differentiate the two multiplication operations.
5. In what real-world physics scenarios is the concept of a scalar product essential?
The scalar product is fundamental for defining several key physical concepts where the component of a vector in a certain direction is important. Prime examples include:
- Calculating Work Done: Work is a scalar quantity calculated as the dot product of the Force vector and the Displacement vector (W = F · d).
- Determining Instantaneous Power: Power is the dot product of the Force vector and the Velocity vector (P = F · v).
- Finding Electric or Magnetic Flux: Flux is the dot product of the field vector (Electric or Magnetic) and the area vector (Φ = E · A or Φ = B · A).
6. Can the scalar product of two non-zero vectors be zero? If so, what does this signify?
Yes, the scalar product of two non-zero vectors can be zero. This specific situation occurs when the angle (θ) between the two vectors is exactly 90 degrees, because cos(90°) = 0. This mathematically signifies that the two vectors are perpendicular (or orthogonal) to each other. It is a very common method used in physics and mathematics to test for orthogonality.
7. What are the key properties of a vector product?
The vector product has several unique properties that are important for physics calculations:
- It is anti-commutative, meaning the order of multiplication reverses the direction of the result: A × B = - (B × A).
- It is distributive over addition: A × (B + C) = (A × B) + (A × C).
- The vector product of two parallel or anti-parallel vectors is always a zero vector, because the angle between them is 0° or 180°, and sin(0°) and sin(180°) are both zero.
8. Why does the vector product A × B result in a vector that is perpendicular to both A and B?
This perpendicularity is an intrinsic part of the geometric definition of the cross product. The operation is designed to describe physical quantities that act perpendicularly to a plane of interaction. For example, torque is the rotational effect of a force, and it acts along the axis of rotation, which is perpendicular to both the force vector and the lever arm vector. The right-hand rule provides a consistent way to determine this perpendicular direction, making the cross product a perfect tool for modelling such three-dimensional phenomena.
9. How does the commutative property differ between scalar and vector products, and what is the physical implication?
The difference in their commutative property has significant physical meaning:
- Scalar Product is Commutative: The order does not matter (A · B = B · A). Physically, this implies that the projection of vector A onto B gives a result that is equivalent to the projection of B onto A in terms of scalar magnitude.
- Vector Product is Anti-commutative: The order is crucial (A × B = - (B × A)). The negative sign indicates that reversing the order produces a new vector with the same magnitude but in the exact opposite direction. This is vital for physical quantities like torque or angular momentum, where the direction (e.g., clockwise vs. counter-clockwise rotation) depends directly on the order of operation.
