Introduction
Cross Product of Two Vectors is a concept that comes under Vector Algebra. Vectors are of different kinds, and we can perform various operations on them ranging from addition, subtraction, multiplication.
Here, we will take a look at how we can multiply them and get a cross-product out of it. In simple terms, the method of multiplying two vectors is what we call the Cross Product of Two Vectors.
We donate this cross product by putting the multiplication sign of (×) between the two vectors, from where the term "cross product" comes.
We define this operation in a three-dimensional system.
In Geometrical Terms:
The area of a parallelogram is the cross-product of two vectors. That cross-product is itself a third vector perpendicular to its two original vectors. This cross product is also generally known as a Vector Product as this result is itself a vector quantity.
Now let us discuss a Cross Product of Two Vectors in detail.
Cross Product of Two Vectors
The cross vector product, area product, or the vector product of two vectors can be defined as a binary operation on two vectors in three-dimensional (3D) spaces. It can be denoted by ×. The cross vector product is always equal to a vector.
Cross Product is a form of vector multiplication that happens when we multiply two vectors of different types. A vector is something that has a direction and a magnitude in nature.
When we do these multiplications, one thing to note is that the product of two vectors is also a vector quantity. In other words, the cross vector product is always equal to a vector.
What is a Vector?
As discussed above, a Vector is an object having both a magnitude and a direction. If we look at this geometrically, we can define a vector as a directed line segment.
The picture given below shows a vector:
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A vector has magnitude (that is the size) and direction.
The vector's direction is from its head to its tail. This line segment's length is the vector's magnitude, and it has an arrow that tells its direction.
Now, we can add two vectors by simply joining them head-to-tail, refer to the diagram given below for better understanding:
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If there are two vectors with the same magnitude and direction, the vector we will obtain would be the same no matter where we change its position. (without rotating the said vector)
It doesn't matter in which order the two vectors are added, we get the same result anyway:
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Labeling a Vector
We can write a vector in bold, for example, a or b.
We can also write a vector as the letters that are on the two sides (tail and arrow) of the line.
The Magnitude of the Vector Product
We could be given the magnitude of the vector as:
|c¯| = |a||b|sin θ,
Where a and b are the magnitudes of the vector and θ is equal to the angle between the two given vectors. In this way, we understand that there are two angles between any two given vectors.
These two angles are θ and (360° - θ). When we follow this rule we consider the smaller angle which is less than 180°.
Some More Information about Cross Products
We use the symbol that is a large diagonal cross (×), to represent this operation, that is where the name "cross product" for it comes from. Since this product has magnitude and direction, it is also known as the vector product.
A × B = AB sin θ n̂
The vector n̂ (n hat) is a unit vector perpendicular to the plane formed by the two vectors. The direction of n̂ is determined by the right-hand rule, which will be discussed shortly.
Direction of the Vector Product
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It should be noted that the cross-product of any unit vector with any other will have a magnitude of one. (The sine of 90° is one, after all.) The direction is not intuitively obvious, however. The rule for cross-multiplication relates the direction of the two vectors along with the direction of the product of the two vectors.
Since cross multiplication is not commutative, the order of operations is important. A right-handed coordinate system, which is known to be the usual coordinate system used in mathematics as well as in Physics, is one in which any cyclic product of the three coordinate axes is positive and any anti-cyclic product is negative.
The right-hand thumb rule is used in which we curl up the fingers of the right hand around a line perpendicular to the plane of the vectors a and b and then curl the fingers in the direction from a to b, then the stretched thumb points in the direction of c.
Step 1: You need to hold your right-hand flat with your thumb perpendicular to your fingers but do not bend your thumb at any time.
Step 2: Now you need to point your fingers in the direction of the first given vector.
Step 3: Orient your palm so that when you fold your fingers, your fingers point in the direction of the given second vector.
Step 4: Your thumb now points in the direction of the cross product of the two vectors.
You can imagine a clock with the three letters x-y-z on it instead of the usual numbers.
Any product of these three letters that is x, y, and z that runs around the clock in the same direction as the sequence of the variables x-y-z is cyclic and positive. Any product that runs in the opposite direction is anti-cyclic and is negative.
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Properties of a Cross Product
Commutative Property
Unlike the scalar product, the cross-products are not commutative,
So where for scalar products The formula is:
a.b = b.a
We have this formula for the vector products:
a × b ≠ b × a
Hence, we can conclude that the magnitude of the cross product of vectors a × b and b × a is the same and is donated by absinθ.
However, suppose we use the right-hand curling method in this example. In that case, we will observe that the two vectors will be in opposite directions.
This would turn into:
a × b = - b × a
Distributive Property
The vector product of two vectors is distributive whether we are talking about a scalar product or a vector addition.
Mathematically, a x (b + c) = a x b + a x c
To get a vector product of any of the two vectors, we can calculate that
Just the same way, the unit factors have results that also hold good,
The Cross Product is Distributive
A × (B + C) equals (A × B) + (A × C)
but not commutative…
A × B = −B × A
Reversing the order of cross multiplication reverses the direction of the product. Since two similar vectors tend to produce a degenerate parallelogram with no area, the cross product vectors of any vector with itself is zero, that is A × A is equal to 0. Now, Applying this corollary to the unit vectors means that the cross product vectors of any unit vector with itself are always equal to zero.
î × î = ĵ × ĵ = k̂ × k̂ = (1)(1)(sin 0°) = 0
Cross Product of Two Vector Product Formula
Let u = ai + bj + ck and v = di + ej + fk be vectors then we define the cross product v x w by the determinant of the matrix:
\[\begin{bmatrix} i & j & k\\ a & b & c\\ d & e & f\end{bmatrix}\]
We can compute this determinant as,
\[\begin{bmatrix} b & c\\ e & f \end{bmatrix}\]i - \[\begin{bmatrix} a & c\\ d & f \end{bmatrix}\]j + \[\begin{bmatrix} a & b\\ d & e \end{bmatrix}\]k
Questions to be Solved
Vector Product Example
Question 1: Find the product of the following using vector product formula: u = 2i + j - 3k ,v = 4j + 5k.
Solution: We calculate the product of the two vectors u and v,
\[\begin{bmatrix} i & j & k\\ 2 & 1 & -3\\ 0 & 4 & 5\end{bmatrix}\] = \[\begin{bmatrix} 1 & -3\\ 4 & 5 \end{bmatrix}\]i - \[\begin{bmatrix} 2 & -3\\ 0 & 5 \end{bmatrix}\]j + \[\begin{bmatrix} 2 & 1\\ 0 & 4 \end{bmatrix}\]k
= 17i - 10j + 8k
This is all about the topic cross product of two vector quantities. Learn how this product is being conducted by following a particular process to determine the outcomes.
FAQs on Cross Vector Product of Two Vectors
1. What is the Vector Product of Two Vectors?
The vector product of two vectors basically refers to a vector that is perpendicular to both of the vectors. One can obtain its magnitude by multiplying their magnitudes by the sine of the angle that exists between them
2. What is Cross-product and Why is the Cross Product of Two Vectors not Commutative?
The vector n will be shown by the thumb. The thumb will show the direction of the vector. The direction of a×b will not be the same as b×a. Thus, the cross product of two given vectors does not obey the commutative law.
3. Is Cross Product a Vector or a Scalar?
Talking about the multiplication of two vectors, one can do two different types of operations. These operations will give us different types of results.
The first type, Dot Product, is a product that we get from two vectors: a scalar product.
The second type of product we get by multiplying two vectors is the Cross Product, where we get a vector product.
The difference is that in the Dot Product, the cross-product of two vectors contains essential information about the said two vectors themselves.
4. What are the different kinds of vectors in Geometry?
The two rudimentary and vital topics in maths and physics are scalars and vectors. They both are different as Scalars are quantities that only have a magnitude value.
In contrast, Vector Quantities are the ones that contain a magnitude and a direction. Vectors in this sense are of different types and kinds, the ten different types of vectors in mathematics are:
Unit Vector
Equal Vector
Zero Vector
Negative of a Vector
Collinear Vector
Displacement Vector
Coplanar Vector
Like and Unlike Vector
Position Vector
Co-initial Vector
These vectors are essential and valuable for further, advanced-level scientific solutions.
5. Which symbol do we use to donate Cross Product?
Geometrically speaking, we can find the cross-product of two vectors by measuring the area of the parallelogram between them.
By measuring this product, we often end up developing a large diagonal cross.
That's why we use the (×) symbol, often referred to as the large diagonal cross, to represent this operation, the term "cross product" from this concept.
As this product also has a direction and a magnitude, this product can be considered a vector term. Hence, we can also call it a Vector Product.
6. What is the Right-Hand Thumb Rule? How is it used?
The right-hand thumb rule is something we use to identify the direction of the cross vector,
When we curl up the fingers of our right hand, making it perpendicular to the plane where the vectors a and b are lying.
After this, we curl the fingers in the direction of the line of a to b. Doing this makes our stretched thumb go towards the direction of c.
To do this, you hold your right thumb perpendicular to your other fingers, keeping the thumb and fingers precisely straight.
Now you have to point your fingers in the direction of the first given vector.
After this, orient your palm's direction in such a way that when you're folding your fingers, they point towards the direction of the given second vector.
This will result in your thumb pointing in the direction of the cross product of the two given vectors.
7. What is the concept of the Right-Handed Coordinate System?
Often referred to as the Usual Coordinate System when used in mathematics as well as in physics, a right-handed coordinate system is the calculation process where a cyclic product of given three coordinate axes is positive, whereas, any anti-cyclic product is negative.
We can understand this concept with the help of a clock with three letters, x-y-z instead of typical numbers.
Now any product of these three given letters that are x, y, z will become positive if used in a formula like x-y-z which is a coordinate series of numbers. Any sequence that is anti-cyclic and runs in the opposite direction would be identified to be negative.
In simple terms, the cross product of a cyclic pair is positive and the anti-cyclic pair is negative, even if they are unit vectors.
