Cross Vector Product of Two Vectors

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What is a Vector?

The picture given below shows a vector:

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A vector has magnitude (that is the size) and direction:

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The length of the line or the arrow given above shows its magnitude and the arrowhead points in the direction.

Now, we can add two vectors by simply joining them head-to-tail, refer the diagram given below for better understanding:

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And it doesn't matter in which order the two vectors are added, we get the same result anyway:

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Notation:

A vector can often be written in bold, like a or b.

A vector can also be written as the letters of its tail and head with an arrow above it, as shown on the right side:

 

What is Cross-Product 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.

 

The Magnitude of the Vector Product

The magnitude of the vector product can be given as,

|\[\bar{c}\]| = |a||b|sin θ,

Where a and b can be known as the magnitudes of the vector and θ is equal to the angle between these two vectors. From the figure given below, we can see that there are two angles between any two vectors, that is, θ and (360° – θ). In this rule, we always consider the smaller angle that is less than 180°.

 

Direction of the Vector Product

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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.

 

Commutative Property

Unlike the scalar product, the cross product vectors is not commutative in nature.

Mathematically, for scalar products a.b = b.a 

But for vector products, vector product formula is:-

 

a x b ≠ b x a

 

As we know, that the magnitude of both the cross product of vectors that is a × b and b × a is the same and is given by absinθ; but the curling of the right-hand fingers in case of a × b is from a to b, whereas in case of (b × a) it is from b to a, as per which, the two vectors are said to be in opposite directions.

Mathematically, a x b = - b x a  

 

Distributive Property

Like the scalar product, the vector product of two vectors is also distributive with respect to vector addition.

Mathematically, a x (b + c) = a x b + a x c   

In order to deal with the vector product of any two vectors, we need to calculate the vector product of any two elementary vectors.

 

\[\bar{a}\] x \[\bar{a}\] = 0, as |a||a| sin0⁰

 

Similarly, for the unit vectors following results hold good,

 

\[\hat{i}\] x \[\hat{i}\] = \[\hat{j}\] x \[\hat{j}\] = \[\hat{k}\] x \[\hat{k}\] = 0 and \[\hat{i}\] x \[\hat{j}\] = \[\hat{k}\] 

 

A Little More about the Cross Product!

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.

 

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 which means that the cross product vectors of any unit vector with itself is always equal to zero.

î × î = ĵ × ĵ = k̂ × k̂ = (1)(1)(sin 0°) = 0

 

The 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. 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:

  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.

  2. Now you need to point your fingers in the direction of the first given vector.

  3. Orient your palm so that when you fold your fingers, your fingers point in the direction of the given second vector.

  4. Your thumb now points in the direction of the cross product of the two vectors.

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.  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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The cross product of an cyclic pair of unit vectors is positive.

The cross product of an anti cyclic pair of unit vectors is negative.


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

FAQ (Frequently Asked Questions)

Q1: What is the Vector Product of Two Vectors?

Answer: 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

Q2: What is Cross-product and Why is the Cross Product of Two Vectors not Commutative?

Answer: 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.

Q3: Is Cross-product Scalar or Vector?

Answer: Dot Product. The Cross Product gives a vector answer and is sometimes known as the vector product. But there is also the Dot Product which generally gives a scalar (ordinary number) answer, and is sometimes known as the scalar product.

Q4: What is a Vector Product and What is a Vector Product Example?

Answer: Definition of vector product and a vector product example : a vector c whose length is the product of the lengths of two vectors a and b and the sine of their included angle, whose direction is perpendicular to the plane, and the direction is that in which a right-handed screw rotated from a towards b along the axis c would move.