MAT 1236 12.3 Handout
The Dot Product a  b  a1 , a2 , a3  b1 , b2 , b3  a1b1  a2b2  a3b3
Example 1
(a) 1,0, 1  0,1, 1 
(b)  2i  3 j  k   i  k  
Dot product and Length
a  a  a1 , a2 , a3  a1 , a2 , a3 
Properties
Geometric Meaning For 0     ,
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Formula cos  
a b
a b
Example 2 a  1, 1,1 , b  1, 1, 2
cos  
a b

a b
Orthogonal Vectors Two vectors are orthogonal if the angle between them is a right
angle.
Example 3 Find the value of x such that the given vectors are orthogonal.
a  x, x, 1 , b  1, x,6
2
Projection
Vector Projection of b onto a
proja b 
a b
a
2
a
Scalar Projection of b onto a
compab 
a b
a
Q: How do I know proja b is in the same direction of a ?
Q: Does the length of the vector proja b agree with what we know?
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Proof
A unit vector in the direction of x and a is
So, according to the formula from the last
slide, we have

x x

We know x  compab 



a b
. Thus,
a
x
Example 4 Find the vector projection of b onto a.
a  2,3 , b  4,1
proja b 
a b
a
2
a
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Classwork
1. Find a unit vector that is orthogonal to both i  j and 2i  j  k .
Solution Suppose the required unit vectors are of the form  i   j   k for some
numbers  ,  , and  .
Step 1 Write down the condition for which  i   j   k is orthogonal to i  j .
 i   j   k    i  j   0
0
Step 2 Write down the condition for which  i   j   k is orthogonal to 2i  j  k .
Step 3 Write down the condition for which  i   j   k is a unit vector.
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Step 4 Solve the 3 equations in the above steps for  ,  , and  . (Be sure to write
down necessary connection statements.)
Answers 1 
3
3
3
3
3
3
i
j
k
i
j
k OR
3
3
3
3
3
3
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