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On the other hand, Scalars only has magnitude; a vector combines both of these attributes. |a| or denotes the importance of a vector product. It’s a scalar that’s not negative.
Equality of Vectors
Two vectors a and b are equally written as a = b if they have (i) the same length (ii) the
same or parallel support and (iii) the same sense.
Vector Types
Zero or Null Vector A vector whose terminal and initial points are coincident refers to a zero/ null vector. It is expressed by 0.
A unit vector has a magnitude of one and is denoted by the letter n.
Vectors for free A vectors are free if their beginning point is not defined.
A Vector’s Negative The negative of a, indicated by —a, is a vector with the same magnitude as a given vector a but in the opposite direction.
Similarities and Dissimilarities Vectors When vectors have the same direction, they are like, and unlike when they show opposite direction.
Vectors that are parallel or collinear: Collinear vectors have the same or parallel supports.
Initial Vectors (Vectors): Coinitial vectors have the same starting point.
Vectors that are coterminous: are those that have the same termination point.
Vectors with a Specific Location: A localized vector is a vector that is drawn parallel to another vector through a specific location in space.
Coplanar vectors: If the supports of a system of vectors are parallel to the same plane, it is coplanar. non-coplanar vectors are the opposite of coplanar vectors.
Vector product examples and Physical Representation of Vectors
Students should be aware that the right-hand rule may be used to determine the cross-product of two vectors. For those who are unfamiliar with the right-hand rule, it is simply the resultant of any two vectors. Both of these vectors should be perpendicular to the other two. The magnitude of the final consequent vector can also be determined using the cross-product. Let’s check out the Physical Representation of Vectors.
If you have two vectors, a and b, then the vector product of a and b is c.
c = a × b
As a result, the magnitude of c = ab sin, where is the angle between a and b and the direction of c is perpendicular to both a and b. What should these cross-direction products be now? So we utilize a rule known as the “right-hand thumb rule” to determine the direction.
Let’s say we’re trying to figure out the direction of a and b. We’ll curl our fingers from a to b. So, if we bend our fingers in the direction illustrated in the diagram, our thumb will point in the direction of c, which is upward. This thumb indicates the orientation of the cross product.
Students should keep in mind that the cross product of two vectors, commonly known as the vector product, is indicated as A B. In addition, the resulting vector will be perpendicular to both the A and B vectors.
A few crucial factors that a learner should keep in mind when dealing with vectors. We’ve compiled an essential points list, which you can find below.
A vector quantity will always come from the cross-product of two given vectors.
If the learner changes the order of the vectors in the vector product idea, the resultant vector will have a – sign.
Directions of A and B both are always upright to the plan encompassing them.
The cross-product value of given 2 linear vectors is the null vector.
The Cross Products Formula
We will look into a few formulae on relevant vectors. Let’s start with the cross-product recipe. If we suppose that is the angle formed by any 2 given vectors, we can write the formula as follows:
A . B = AB sin θ
Alternatively,
A × B = ab sin θ n̂
The unit vector is n in this case.
As we’ve already seen, the cross-product of the given two vectors may be stated in the form of a matrix, commonly known as the determinant form. This idiom is demonstrated in the following example.
X Y = I (yc – zb) – j (xc – za) + k X Y = I (yc – zb) – j (xc – za) + k X Y = I (yc (xb – ya)
The triple cross product is the next essential issue after the cross product of two vectors. Further, the triple product is the product value of 3 vectors. Alternatively, it may be described as a vector’s cross product with the cross product of provided 2 additional vectors.
Vector Product of Unit Vectors
The three unit vectors are i^ , j^ and k^. So,
i^ × i^ = 0
i^ × j^ = 1 k^
i^ ×k^ =1 – j^
j^ ×i^ = – k^
j^ ×j^ = 0
j^ ×k^ = 1 i^
k^× i^= j^
k^× j^= -i^
k^× k^= 0
This is how we determine the vector product formula of unit vectors.
We must suppose three vectors, exemplified by A, B, and C, to arrive at the vector product formula for the triple cross product. The following are the symbols for the provided three vectors.
(A. C) B – (A. B) C = (A. C) B – (A. B) C
(A B) C = -C (A B) = -(C. B) A + (C. A) B. (A B) = -(C. B) A + (C. A) B.
Conclusion
Here we learned about vector products and how to calculate them. Make sure to explore the definition in detail and solve examples using the formula for vector products. The article also covers the part of physical representation of the vectors for learners to understand the concept.
