1. Inner Product, Length and Orthogonality
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

2. INNER PRODUCT
For vectors u and v in ℝn we can define their INNER PRODUCT.
This is also called the “dot product”. We have been using dot products to do matrix arithmetic, so it should be a familiar computation.
Some properties of inner products:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

3. LENGTH of a vector
For a vector v in ℝn we can define the LENGTH (or NORM) of v.
If v is a vector in ℝ2 you should recognize this as the length of the hypotenuse of a right triangle (i.e. the Pythagorean Theorem)
Note that if a vector is multiplied by a constant, then its length is multiplied by the same constant:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

4. LENGTH of a vector A vector with length=1 is called a UNIT VECTOR.
Any vector can be turned into a unit vector by dividing by its length.
u is a vector that points the same direction as v, but has length=1.
When we create a unit vector in this way we say that vector v has been “normalized”.
The DISTANCE between vectors u and v is the length of their difference:
This should coincide with the usual “distance formula” that you know for finding the distance between 2 points in ℝ2.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

5. u and v are orthogonal when u•v=0
ORTHOGONALITY
We say that vectors u and v in ℝn are ORTHOGONAL if their inner product is 0.
When two vectors are perpendicular (the angle between them is 90°) we can also call them “orthogonal”. Just a new word for a familiar property.
u and v are orthogonal when u•v=0
Here is a formula for the angle between two vectors:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

6. EXAMPLES Given these vectors in ℝ3, find the following:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

7. EXAMPLES 1) Here is the calculation:
Given these vectors in ℝ3, find the following:
1) Here is the calculation:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

8. EXAMPLES 2) Here is the calculation:
Given these vectors in ℝ3, find the following:
2) Here is the calculation:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

9. EXAMPLES 3) Here is the calculation:
Given these vectors in ℝ3, find the following:
3) Here is the calculation:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

10. EXAMPLES 4) Here is the calculation:
Given these vectors in ℝ3, find the following:
4) Here is the calculation:
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB

11. EXAMPLES
Given these vectors in ℝ3, find the following:
5) To get a unit vector, divide each component by the length of the vector:
both of these unit vectors have length=1 (check this!) and point in the same directions as the original vectors.
Prepared by Vince Zaccone
For Campus Learning Assistance Services at UCSB