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An orthogonal unit vector is a unit vector that is orthogonal to the direction of the second and third vectors. Since the second and third vectors are mutually perpendicular, the unit vector is orthogonal to the vector. This unit vector is used for the measurement of the angle between two vectors. Orthogonal vector systems are generated by the projection of a point on a line perpendicular to a plane. These systems are convenient because they are easy to describe mathematically and have a clear geometrical basis.
Orthogonal Unit Vector
A number of vectors that are mutually perpendicular to each other, meaning they form an angle of 90° with a magnitude of one unit with each other, are called orthogonal unit vectors. The dot product of an orthogonal vector is always zero since Cos90 is zero.
Orthogonal unit vectors are vectors that are perpendicular to each other, and they form a set that can be rotated without changing direction. This is useful for transformations such as scaling and rotating that don’t change the direction of a vector.
Properties of orthogonal vectors
An orthogonal unit vector describes a state of a system in which the system is characterised by a direction and a magnitude. The projection of this vector onto the x-axis represents the magnitude of the system, and the projection of the orthogonal unit vector onto the y-axis represents the direction of the system.
Some of the properties of orthogonal vectors are as follows:
Zero vector is orthogonal to every vector since its dot product with any vector is zero.
Any two vectors are orthogonal if their inner product is zero.
Orthogonal vectors always have zero as their dot product and are perpendicular to each other.
The cross product of two orthogonal vectors can never be zero until it is a zero vector. This is because the angle between orthogonal vectors is 90° and Sin90° is 1. Hence, the cross product will be equal to the product of the magnitude of the orthogonal vectors.
In order for vectors to be orthogonal, they should point in the same direction and opposite directions.
A mathematical theorem states that any set of vectors that is orthogonal is also linearly independent.
General equation of orthogonal vectors in 2-D
In 2-D, orthogonal lines are lines that don’t share a common angle. Like parallel lines, two orthogonal lines never intersect.
a.b = 0
(axbx) + (ayby) = 0
(aibi) + (ajbj) = 0
General Equation of orthogonal vectors in 3-D
Orthogonal vectors in 3-D space are used to represent the direction of a surface or the position of an object.
a.b = 0
(axbx) + (ayby) + (azbz) = 0
(aibi) + (ajbj) + (akbk) = 0
Examples of orthogonal vectors
An orthogonal vector is a vector that is perpendicular to two scalar values. In other words, an orthogonal vector is a vector that is at a right angle to another vector. For example, the two vectors in the image on the right are orthogonal because they are at a right angle to each other.
For example, the vector [1,0,0] is the same as [0,1,0]. The vector [1,0,0] is perpendicular to the scalar values, and the two vectors are orthogonal.
Conclusion
Orthogonality is a generalised form of perpendicularity in calculus and linear algebra. Two vectors are said to be orthogonal vectors if their dot product is zero and are perpendicular to each other. If we need to find if the two vectors are orthogonal or not, we simply take their dot product and check whether or not it is equal to zero. In mathematical terms, orthogonal vectors are those whose transpose is equal to their inverse. Orthogonality in vectors is the linear transformation of vector equations that preserve the length of the vector.
