Unit Vectors

A vector has both magnitude and direction. A unit vector is a vector with magnitude of 1. In some situations it is helpful to find a unit vector that has the same direction as a given vector.

 A unit vector of v, in the same direction as v, can be found by dividing v by its magnitude v.

UNIT VECTOR:


If v0 and v represents the magnitude of vector v, then its unit vector u is:



u= v v = 1 v v


The unit vector u has a length of 1 in the same direction as v.



The unit vectors 1,0 and 0,1 are special unit vectors called standard unit vectors and are represented by the vectors i and j as follows:

i=1,0       j=0,1

Any vector in a plane can be written using these standard unit vectors.

v= v 1 , v 2 = v 1 i+ v 2 jLinear Combination


This vector sum is called a linear combination. For example, vector v=3,11=3i+11j .

Let's look at some examples.

To work these examples requires the use of various differentiation rules. If you are not familiar with a rule go to the associated topic for a review.


Example 1: Find a unit vector u in the same direction as v=12,9 and show that it has a magnitude of 1.

Step 1: Find the magnitude of v.


The magnitude of a vector is calculated by taking the square root of the sum of the squares of its components.


v= x,y


v= x 2 + y 2

v=12,9


||v|| = 12 2 + ( 9 ) 2


v= 225 =15

Step 2: Calculate the unit vector.


u= v v = 1 v v

u= v v = 1 v v


u= 12,9 15 = 1 15 12,9


u= 12 15 , 9 15 = 4 5 , 3 5

Step 3: Show that the vector u has a magnitude of 1.


The magnitude of a vector is calculated by taking the square root of the sum of the squares of its components.


v= x,y


v= x 2 + y 2

u= 4 5 , 3 5


||u|| = ( 4 5 ) 2 + ( 3 5 ) 2  


||u|| = 16 25 + 9 25   = 25 25


u= 1 =1

Example 2: If u = -3i + 2j and v = -i + 6j, find 2u + 4v.

Step 1: Find vectors 2u and 4v using scalar multiplication.


kv=k v 1 , v 2 =k v 1 ,k v 2 Scalar Multiplication

2u=2( 3i+2j )=[ 2 ·( 3i )+2 ·2j ]


2u=6i+4j


4v=4( i+6j )=[ 4 ·( i )+4 ·6j ]


4v=4i+24j

Step 2: Add vectors 2u and 4v using vector addition.


u+v= u 1 + v 1 ,  u 2 + v 2 Vector Addition

2u + 4v


(-6i + 4j) + (-4i + 24j)


(-6i - 4i) + (4j + 24j)


( 10i+28j )





Related Links:
Math
algebra
Direction Angles of Vectors
The Dot Product
Pre Calculus


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