A unit vector is a vector that has a magnitude (length) equal to 1. It is used primarily to indicate direction without regard to magnitude. Unit vectors are often denoted by a lowercase letter with a "hat" symbol, for example, v^\hat{v}v^ (pronounced "v-hat")
Key Points About Unit Vectors:
- The magnitude of a unit vector is exactly 1.
- Unit vectors have the same direction as the original vector but are scaled to length 1.
- They are often used to represent directions, such as normal directions to surfaces.
- Commonly used unit vectors in three-dimensional space are i^\hat{i}i^, j^\hat{j}j^, and k^\hat{k}k^, which point along the x, y, and z axes respectively, each having magnitude 1
How to Find a Unit Vector:
To find the unit vector A^\hat{A}A^ in the direction of a given nonzero vector AAA, divide the vector by its magnitude ∣A∣|A|∣A∣:
A^=A∣A∣\hat{A}=\frac{A}{|A|}A^=∣A∣A
where ∣A∣=x2+y2+z2|A|=\sqrt{x^2+y^2+z^2}∣A∣=x2+y2+z2 if A=(x,y,z)A=(x,y,z)A=(x,y,z)
Example:
If A=(1,2,3)A=(1,2,3)A=(1,2,3), then its magnitude is ∣A∣=12+22+32=14|A|=\sqrt{1^2+2^2+3^2}=\sqrt{14}∣A∣=12+22+32=14. The unit vector in the direction of AAA is:
A^=(114,214,314)\hat{A}=\left(\frac{1}{\sqrt{14}},\frac{2}{\sqrt{14}},\frac{3}{\sqrt{14}}\right)A^=(141,142,143)
Unit vectors are fundamental in vector spaces as they form bases for expressing any vector as a linear combination of unit vectors
. In summary, a unit vector is a normalized vector used to specify direction with a fixed length of 1, making it a crucial concept in mathematics, physics, and engineering.
