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Equality of Vectors
Two vectors a and b are said to be equally written as a = b if they have (i) the same length, (ii) the same or parallel direction/support, and (iii) the same sense.
Types of Vectors
- Zero or Null Vector: A vector whose initial and terminal points are coincident is called a zero or null vector. It is denoted by 0
- A unit vector has a magnitude of one and is denoted by the letter n
- Free Vectors: A vector is free if its beginning point is not defined
- A Vector’s Negative: The negative of vector a, indicated by , is a vector with the same magnitude as a given vector a but in the opposite direction
- Similarities and Dissimilarities in Vectors: When vectors have the same direction, they are said to be like, and when they have the opposite direction, they are said to be unlike
- Vectors that are parallel or collinear: Collinear vectors have the same or similar supports
- Initial Vectors/Coinitial Vectors have the same starting point
- Vectors that are coterminous: Coterminous vectors are those that have the same termination point
- Vectors with a Specific Location: A localised 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
Students should be aware that they may use the right-hand rule 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.
If you have two vectors, a and b, then the vector product of a and b is c.
As a result, the magnitude of , where theta θ 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 utilise 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 b. If we curl our fingers from a to b, our thumb will point in the direction of c, which is upward. This thumb indicates the orientation of the cross product.
Students should remember 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 consider when dealing with vectors. We’ve compiled a list of those essential points below.
- A vector quantity will always come from the cross-product of any two vectors
- If the learner changes the order of the vectors in the vector product idea, the resultant vector will have a negative sign
- Both A and B’s directions will always be perpendicular to the plan that encompasses them
- The cross-product of any two linear vectors is the null vector
The Formula of Cross Products
There are a few formulae in the chapter on relevant vectors. We’ll look at some essential vector formulae in this part. Let’s start with the cross-product formula. If we suppose that θ is the angle formed by any two given vectors, then the formula is as follows:
A . B = AB cos θ
Alternatively, the same formula can be written as
A × B = AB sin θ n̂
The unit vector is n in this case.
As we’ve already seen, the cross-product of these two vectors may be stated in the matrix form, commonly known as the determinant form. This idiom is demonstrated in the following example.
X×Y = i (yc – zb) – j (xc – za) + k (xb – ya)
The triple cross product is the next essential issue after the cross product of two vectors. As you might have guessed, the triple product is the product of three vectors. Alternatively, it may be described as a vector’s cross product with the cross product of any two additional vectors.
Vector Product of Unit Vectors
The three unit vectors are i^ , j^ and k^. So,
- i^ × i^ = 0
- i^ × j^ = k^
- i^ ×k^ = – j^
- j^ ×i^ = – k^
- j^ ×j^ = 0
- j^ ×k^ = i^
- k^× i^= j^
- k^× j^= -i^
- k^× k^= 0
This is how we determine the vector product formula of unit vectors.
Conclusion
We must suppose that there are three vectors, represented by A, B, and C, to arrive at the vector product formula for the triple cross product. The following are the symbols for these three vectors.
A×(B×C) = (A. C) B – (A. B) C
(A×B)×C = (A.C)B – (B.C)A
A×(B×C) ≠ (A×B)×C
