1. 8.6.1 – The Dot Product (Inner Product)

2. So far, we have covered basic operations of vectors – Addition/Subtraction – Multiplication of scalars – Writing vectors in various forms We will now talk about the last crucial operation

3. Dot Product The product of two vectors will create a scalar The dot product of two vectors is given if u = {u 1, u 2 } and v = {v 1, v 2 } The dot product may be positive, negative, or zero (similar to multiplication of real numbers)

4. Example. Find the dot product if u = {-5, 2} and v = {3, -1} Find each corresponding part

5. Example. Find the dot product if u = {-5, 2} and v = {-5, 2}

6. Example. Find the dot product if u = {-5,2} and v = {2, 5}

7. Properties With the dot product, we can derive certain properties 1) u. v = v. u (commutative) 2) 0. u = 0 3) u. (v + w) = u. v + u. w (distribution) 4) a(u. v) = (au). v = u. (av) 5) u. u = ||u|| 2

8. Example. Find the quantity 3v. u if u = {-2, 3} and v = {4, 4}

9. Example. Find the magnitude of the vector v if the dot product with itself is 12.

10. Example. u. u = 80. Find ||u||.

11. Dot Product Theorem Similar to component form, we can talk about the dot product of vectors in terms of an angle Let u and v be nonzero vectors, and ϴ be the smaller of the two angles formed by u and v; then,

12. Example. Find the angle between the two vectors u = {5,4} and v = {3, 2}

13. Example. Find the angle between the two vectors u = 5i + 2j, v = 4i + j

14. Assignment Pg. 678 1-23 odd