How is a unit vector calculated?

How is a unit vector calculated?

• A. By multiplying the vector by its magnitude
• B. By dividing the vector by its magnitude
• C. By adding the components of the vector
• D. By taking the cross product of the vector

Answer: (B)

A unit vector is defined as a vector that has a magnitude of one, which is crucial for normalizing vectors in various applications such as physics and engineering. To compute a unit vector from an original vector, you divide each component of the vector by its magnitude. This process effectively scales the vector such that its length is one, while maintaining its direction. Calculating the magnitude of a vector involves taking the square root of the sum of the squares of its components, following the Pythagorean theorem in multiple dimensions. Once you have the magnitude, dividing each component of the vector by this magnitude gives you the corresponding components of the unit vector, resulting in a new vector with a length of one. Other methods listed—such as multiplying the vector by its magnitude, adding the components of the vector, or taking the cross product—do not yield a unit vector. Multiplying the vector by its magnitude would increase its length rather than normalize it, while simply adding components would not provide a directionally meaningful result. The cross product serves a different purpose entirely, producing a new vector perpendicular to the original vectors involved, rather than a unit vector in the same direction.

A unit vector is defined as a vector that has a magnitude of one, which is crucial for normalizing vectors in various applications such as physics and engineering. To compute a unit vector from an original vector, you divide each component of the vector by its magnitude. This process effectively scales the vector such that its length is one, while maintaining its direction.

Calculating the magnitude of a vector involves taking the square root of the sum of the squares of its components, following the Pythagorean theorem in multiple dimensions. Once you have the magnitude, dividing each component of the vector by this magnitude gives you the corresponding components of the unit vector, resulting in a new vector with a length of one.

Other methods listed—such as multiplying the vector by its magnitude, adding the components of the vector, or taking the cross product—do not yield a unit vector. Multiplying the vector by its magnitude would increase its length rather than normalize it, while simply adding components would not provide a directionally meaningful result. The cross product serves a different purpose entirely, producing a new vector perpendicular to the original vectors involved, rather than a unit vector in the same direction.