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About Vector cross product
Introduction
Vector Product is also known as cross vector or cross product, it can be defined as the way by which two vectors are combined or grouped together is known as the vector product i.e., when we find out the vector product of two or more vectors, it is named as vector Product or cross product.
Vector product is also known as a cross-product because at the time of finding a vector product we use “x” as the sign to find out the vector. The sign is pre-defined and shows that the product of two variables is done. Some people or textbooks use the ‘wedge’ symbol “^” as a sign to find out the vector product of two variables. But, both the signs are correct and can be used to determine the vector product of variables.
Methods to Find out Vector Product of two variables
There are mainly two ways (Right-hand thumb rule) by which a vector product of two vectors can be found or obtained. These are-
The first method is to keep the first finger of the right hand named as ‘a’ and then align the middle finger of the right hand named as ‘b’. Also, align your thumb too. If these two fingers i.e., ‘a’ and ‘b’, and your thumb are making the right angle or positioned at the right angle then the thumb shows or points towards the direction of ‘a’ and ‘b’. Hence, the result is a x b.
The second method to find out the direction of two vectors is also determined by the right-hand thumb rule. It can be found out by moving your fingers i.e., ‘a’ and ‘b’ in the direction in which a finger named as ‘a’ would be rotated to meet finger ‘b’. Then, the thumb points in the direction of a x b, which shows the product of two vectors i.e., ‘a’ and ‘b’.
Properties of the vector product
Vector Product is not Communicative
The Vector product of two or more variables is not communicative means in other words we can say that the direction of two or more variables does not affect the calculation of the vector product means it does not matter whether b x a is in the opposite direction of a x b. Let two variables be ‘a’ and ‘b’, so according to this property, b x a = -a x b.
Vector Product is distributive Over Addition
The other property of a vector product is that it is distributive over addition. In simple words, we can say that if we are doing a product with other two variables then it is distributive to them over addition. This means a x (b + c) = a x b + a x c. Hence, a x (b + c) = a x b + a x c is always equal to (b + c) x a = b x a + c x a.
Vector Product of two Vectors Parallel vectors is Zero
The next property of a vector product is that the vector product of two parallel vectors is zero, that is the result is always zero at last. , means a x b = 0 or b x a = 0.
For Example:
Let ‘a’ and ‘b’ be two vectors and these are parallel to each other. Here, the definition of a vector product Is not applied because ‘a’ and ‘b’ are parallel which does not define a plane. So, here the vector ‘n’ (let's suppose) is not perpendicular to the plane. Hence, when two vectors are parallel and we want to find out their vector product, the vector product is always zero. This means for two parallel vectors, a x b = 0.
Note
Vector product of two vectors given in Cartesian form has many properties or we can say that-
Let ‘a’, ‘b’, and c are vectors, which define the following properties on the Cartesian plane. a × b = 0 b × b = 0 c × c = 0 a × b = c b × c = a c × a = b b × a = −c c × b = −a a × c = −b
applications of the vector product
Here, we will know where the vector product is mostly applicable or mostly used by people. The applications of vector product or cross vector are-
The first application of a vector product is to find out the perpendicular of two given vector variables. It is one of the most common applications which is used everywhere to find out the vectors which are perpendicular to the given vector. The point that is kept in mind while finding the perpendicular of two vectors is that the two vectors should not be parallel and they should be non-zero, which means any number but not zero.
The next most common application of using vector product is to find out the area of the parallelogram means the length of the base of the parallelogram is multiplied (product) by the perpendicular height of the parallelogram. In other words, we can say that, sin θ = h |c| and so h = |c| sin θ. Hence, the area of the parallelogram is Area = |b × c|.
The next application of using a vector product of two vectors is to find out the volume of the parallelepiped.
What is vector cross product?
Vector cross product is the the way to find the vector which is perpedicular to a plane surface.
what is Vector cross product formula?
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)
What is the application of vector cross product calculator ?
following are the application of cross product calculator
- Vector cross product calculator is used to get the vector perpendicular to the plane
- It is used to know the angle between two plane
- cross product calculator is used to get the the area of plane
what is unit vector cross product ?
Geometrically, the cross product of two vectors is the area of the parallelogram between them. The symbol used to represent this operation is a large diagonal cross (×), which is where the name "cross product" comes from. Since this product has magnitude and direction, it is also known as the vector product .