The Dot Product

The dot product of two vectors is the product of the two x-components plus the product of the two y-components.

DOT PRODUCT OF VECTORS:


Let u= u 1 , u 2 and v= v 1 , v 2


u·v= u 1 v 1 + u 2 v 2




The vector operations of scalar multiplications and vector addition both result in a vector. However, the dot product results in a scalar or a number, not a vector.

The following are properties of the dot product.

PROPERTIES OF THE DOT PRODUCT:

Let u,v and w be vectors and c be scalars.


1. u · v = v · u      2. 0 · v = 0

3. u · (v · w) = u · v + u · w4. v · v = ||v||2

5. c(u · v) = cu · v = u · cv


Let's look at some examples.

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


Example 1:     Let u=5,7 , v=3,1 and w=-6,-4 . Find each dot product:
     (a) u · v     (b) u · w     (c) v · w

Step 1: Evaluate u · v.


Remember the result will be a scalar.

u·v= u 1 v 1 + u 2 v 2


( 5·3 )+( 7·1 )


15+7=22

Step 2: Evaluate u · w.


Remember the result will be a scalar.

u·w= u 1 w 1 + u 2 w 2


( 5·6 )+( 7·4 )


3028=58

Step 3: Evaluate v · w.


Remember the result will be a scalar.

v·w= v 1 w 1 + v 2 w 2


( 3·6 )+( 1·4 )


184=22

Example 2:     Let u=2,1 , v=10,-3 and w=4,5 . Find each dot product:
     (a) (v · w) · u     (b) 2v · w     (c) (u + v) · w

Step 1: Evaluate (v · w) · u using Property 3.


Remember the dot product will be a scalar.

( v ·w )u=u·v+u·w


( u 1 v 1 + u 2 v 2 )+( u 1 w 1 + u 2 w 2 )


[ ( 2·10 )+( 1·3 ) ]+[ ( 2·4)+(1·5 ) ]


17+13=30

Step 2: Evaluate 2v · w.


First multiply v by 2 using scalar multiplication and then find the dot product, which will be a scalar result.

2v ·w=210,3·4,5


20,6·4,5


( 20·4 )+( 6·5 )


8030=50

Step 3: Evaluate (u + v) · w.


First add vectors u and v using scalar multiplication then find the dot product of the resultant vector and w. Remember the dot product will be a scalar.

( u+v )·w


[ u 1 + v 1 , u 2 + v 2 ]·w


[ ( 2+10 ),( 1+( 3 ) ) ]·w


( 12,2 )·w


( u+v )·w= x 1 w 1 + y 2 w 2


( 12·4 )+( 2·5 )


48+( 10 )=38





Related Links:
Math
algebra
Angle between Two Vectors
Decomposing a Vector into Components
Pre Calculus


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