Unit Vector

Unit Vector

Definition and Basic Properties:

• A Unit Vector is a vector with a magnitude (length) of 1.
• It is represented with a caret (^) on top of the vector notation such as â or û.
• A unit vector always maintains the direction of the original vector but its magnitude is set to 1.
• Another way of describing a unit vector is that it is a direction vector.

Conversion of Vector to a Unit Vector:

• Any non-zero vector v can be converted into a unit vector û in the same direction by dividing the vector by its magnitude.

• The formula is **û = v /

  v

  **, where **

  v

  ** is the magnitude of vector v.

• Before a vector can be converted to a unit vector, its magnitude must be calculated.

Unit Vectors in Cartesian Plane:

• In a 2D Cartesian plane, there are three notable unit vectors, namely â along x-axis, b̂ along y-axis, and û in any chosen direction.
• Similarly, in a 3D Cartesian system, there are three unit vectors corresponding to each of the three dimensions. These are î along the x-axis, ĵ along the y-axis, and k̂ along the z-axis.
• In terms of these unit vectors, any arbitrary vector v can be expressed as v = xâ + yb̂ in 2D and v = xî + yĵ + zk̂ in 3D.

Practical Applications of Unit Vectors:

• Unit vectors are useful in physics and engineering to represent directions.
• They simplify calculations as their magnitude is always 1.
• They are often used in geometry and trigonometry to represent angles.

Important Reminders on Unit Vectors:

• Multiplying a unit vector by a scalar changes its magnitude but not its direction.
• Adding two unit vectors results in a new vector whose magnitude is not necessarily 1.
• Any non-zero vector divided by its own magnitude results in a unit vector.
• The concept of unit vectors is fundamental to vector algebra and vector calculus.