Cross Vector Product of Two Vectors

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.