A vector, demonstrated by is determined by two points A, B such that the magnitude of the vector is the length of the straight line AB and its direction is from A to B. Point A is called the initial point of the vector, and B is called the terminal point.
The magnitude or modulus of a vector can be seen as a positive number that measures its length and is denoted as the modulus of a vector is sometimes written as a.
Now, different types of operations can be performed on these vectors, such as addition, subtraction, multiplication, division, etc.
We will see how to do the cross-product of vectors, their formula, and their properties with some examples through this article.
What is Cross Product?
Cross product can be specified as the multiplication of two vectors. A cross product is represented by the multiplication sign (in between two vectors. Cross product can also be seen as a binary vector operating in a three-dimensional system.
If a cross product of two vectors is taken then it will always give a third vector that is perpendicular to the taken vectors. We also have many online cross-product calculators where you can find the cross-product between two vectors within the seconds.
A cross product is sometimes used to determine the vector perpendicular to the plane surface traversing the two vectors. The cross product of two vectors let’s say a, b, is equal to another vector that is at right angles to both the vectors, and it occurs in three dimensions.
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If A and B are taken as two independent vectors, then the cross product of these two vectors (AB) will be perpendicular to both vectors, and it will be normal to the plane having both vectors.
The critical thing to remember is that the result is a vector and NOT a scalar value. This is why it is known as a vector product. For the definition easy to remember, we generally use determinants to calculate cross-products.
a ⨯ b = |a| |b| sin (θ) n
- | a | and | b | – Length of two vectors.
- θ – Angle between the two vectors a and b.
- n – Unit vector perpendicular to both vectors a and b.
Cross Product For Two Vectors
The formula for the cross-product of two vectors can be derived by the following method.
Let’s take two vectors as
A = ai + bj + ck
B= xi + yj + zk
We know that i, j and k are standard basis vectors that have below-given equalities.
i j = k and j i = -k
j k = i and k j = -i
k i = j and i k = -j
i ×i = j × j = k × k = 0
A × B = (ai + bj + ck) × ( xi + yj + zk)
= ax(i × i) + ay(i ×j)+ az(i × k) + bx(j × i) + by(j × j) + bz(j × k) + cx(k × i) + cy(k × j) + cz(k × k)
Apply the vectors as mentioned above equalities-
A × B = ax(0) + ay(k) + az(-j) + bx(-k) + by(0) + bz(i) + cx(j) + cy(-i) + cz(0) = (bz – cy)i + (cx – az)j + (ay – bx)k
Here, note that i × j j × i; thus, the cross product is not commutative, and the associative law does not hold. (a × b) × c a × (b × c)
Example: Find the cross product of the below given two vectors
= (3, 4, 5) and = (7, 8, 9)
The cross product is given as-
a = 3i + 4j + 5k
b = 7i + 8j + 9k ;
a×b = i – j + k
= ( 36 – 40)i – (27 – 35)j + (24 – 28)k = -4i + 8j -4k
So, × = -4i + 8j – 4k
Right Hand Rule For Cross Product
We can find the direction of the unit vector by taking into account the right-hand rule. To decide the right cross-product, we have a right-hand rule.
For making use of this rule, you hold your right hand up, then lift your index finger and en route towards the first vector, and now point your middle finger in the direction of the second vector.
While doing this, the thumb of your right hand will show the direction of the unit vector.
By taking into account the right-hand rule, we can easily demonstrate and testify that the cross product of vectors is not commutative.
We denote the cross product between two vectors as “a” cross “b” (a × b ).
Let’s assume that a × b = c.
Here, vector c shows two unique properties.
First, the unit vector here is orthogonal to both a and b. This is also the reason why cross-product only works in three dimensions. Since, in two dimensions, vectors are not always perpendicular to any pair of other vectors.
Second, the length of c shows the measure of how far a and b are pointed and elevated by their magnitudes.
If a, b and c are vectors and c is a scalar then, the properties of cross-product are-
- a × b = -b × a
- (ca) × b = c(a × b) = a ×(cb)
- a × (b + c) = a × b + a × c
- (a + b) × c = a × c + b × c
The length of the cross product a×b, a × b is equal to the area of the parallelogram determined by sides a and b.
Example 1: Find the area of the triangle with the vertices P(0,1,4), Q(-5, 9,2), and R(7, 2, 8)
PQ = (-5, 8,-2) and PR = (7, 1,4)
So, PQ × PR = ( 34, 6, -61)
Area of parallelogram = PQ × PR= (34² +6² +(-61)²)= 17(17)
Thus, the area of the triangle = 17(17)/2 35
The parallelepiped volume calculated by the vectors a, b, and c gives the magnitude of their scalar triple product.
V = a . (b ×c)
If the triple scalar product is 0, then it means that vectors lie in the same plane; that is, they are coplanar.
Example 2: Find the volume of the parallelepiped determined by the vectors a, b, and c.
a = (6, 3,4), b = (0, 2, 1), c = (5, -1, 2)
b×c = (5, 5,-10)
a.(b×c) = (6, 3,-4) . (5, 5, -10) = 85
V= a.(b×c) = 85
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Important Points To Take:
- The cross product of two vectors always shows a vector that is perpendicular or orthogonal to the two vectors.
- The direction of the two vectors in the cross product can be given by the right-hand thumb rule, and the magnitude of the vectors is shown by the area of a parallelogram, which is formed by the original vectors.
- The cross product of two linear or parallel vectors is always a zero vector which is a scalar quantity.
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