Unit Vector

Unit Tangent Vector 

Assuming v(t) is a smooth vector-valued function with a zero-threshold, any vector that is parallel to V‘(t) is considered to be tangent to the graph of v(t) at the time of the zeroth degree. It is most commonly used to represent the direction V’ (t) rather than the magnitude of this same direction.

We are thus examining the unit vector pointing in V ‘(t) direction. In this case, the following is the unit tangent vector definition:

Examine the smooth function V’ (t) on the open interval I, which is defined as follows: Thus, this example denotes the unit tangent vector as t(t).

 t(t) =V‘(t)/|V‘(t)|

Unit Normal Vector 

For simplicity, consider the vector-valued function V (t), in which the unit tangent vector, denoted by the symbol t‘(t), is smooth on the open interval. N (t) is defined as the unit normal vector n (t).

n(t) =t‘(t)/|t‘(t)|

The primary unit vector is another name for this vector.

Applications of Unit Vector

• In Physics, a unit vector is a vector that represents one unit.
• It is a vector in physics with an equal magnitude and a certain direction known as the unit vector. The sole direction is determined by a unit vector, the smallest vector. Dimensions and units are not provided for them.
• The x-axis, y-axis, and z-axis are all represented in a rectangular coordinate system.
• They are parallel and perpendicular to the other unit vectors in different coordinate axes.

Characteristics of Vectors

• It is necessary to comprehend their properties to grasp vectors and execute various computations utilising vectors.
• The dot product of two vectors is a scalar that resides in the plane of the two vectors used in the calculation.
• Crossing two vectors results in a vector perpendicular to the plane in which these two vectors are located.

Points to remember about Unit Vector

• Orthogonal unit vectors are always orthogonal to one another, and their dot product is always zero.
• It is always true that the cross product of parallel unit vectors is zero.
• When the cross product of two or more unit vectors equals zero, they are collinear.
• A non-negative number reflects the magnitude of the vector.

Summary

A vector is a quantity that possesses both magnitude and direction. A vector may be described as a line segment with a specific direction and a particular length from a geometric perspective. Scalars are physical values with just magnitude, such as mass, length, time, temperature, and the distance between two locations. Scalars are used in mathematics to represent the magnitude of physical quantities. It is convenient to portray a vector as a straight line with an arrowhead.