In a two-dimensional coordinate system, the x-component and y-component are commonly considered to be the components of a vector. It can be written as V = (Vx, Vy), with V denoting the vector. The components of vectors created along the axes are these. We’ll use formulas for two-dimensional and three-dimensional coordinate systems to discover the components of any given vector in this article.
Vector items are divided into two groups. Because a vector has both magnitude and direction, the dot product and cross product of two vectors is based on this fact. The dot product of two vectors is also known as a scalar product since the resultant value is a scalar quantity. The cross product is called the product of vectors because the result is a vector that is perpendicular to these two vectors.
Types of Vector Products
The scalar product of vectors is another name for the dot product of vectors. A scalar value is the result of the dot product of the vectors. The dot product of vectors is the product of the magnitudes of two vectors and the cosine of the angle between the two vectors. The dot product of two vectors results in a resultant in the same plane as the two vectors.
|p||q| cos is the scalar product of two vectors of magnitude |p| and |q|, where it represents the angle between the vectors a and b taken in the direction of the vectors.
We can express the scalar product as:
where |p| and |q| denotes the magnitude of the vectors p and q, cos denotes the cosine of the angle between the two vectors, and p.q signifies the dot product of the two vectors, and cos denotes the cosine of the angle between the two vectors.
A Vector Product is another name for a Cross Product. The cross product is a type of vector multiplication in which two vectors of different sorts or natures are multiplied. The resultant vector is called the cross product of two vectors or the vector product when two vectors are multiplied by each other and the product is also a vector quantity. The generated vector is perpendicular to the plane containing the two supplied vectors.
If θ is the angle between the given two vectors I and J, then the formula for the cross product of vectors is given by:
I × J = |I| |J| sin θ
Here are the two vectors and the magnitudes of the given vectors.
θ is the angle between two vectors and is the unit vector perpendicular to the plane containing the given two vectors, in the direction given by the right-hand rule.
Right-hand Rule Cross Product
The direction of the unit vector can be determined using the right-hand rule. We can stretch our right hand in this rule such that the index finger points in the direction of the first vector and the middle finger points in the direction of the second vector. The right hand’s thumb then indicates the direction or unit vector n. Using the right-hand rule, we can easily show that the cross-product of vectors is not commutative.
Cross product of two vectors Formula
Consider two vectors,
A = ai + bj + ck
B = xi + yj + zk
The standard basis vectors m, n, and o meet the equalities shown below.
m × n = o and n × m = –o
n × o = m and o × n = –m
Uses of Product of Vectors
- Projection of a Vector.
- Angle Between Two Vectors.
- Triple Cross Product.
- Area of a Parallelogram.
- The volume of a Parallelepiped.