1. The Dot Product (MAT 170) Sections 6.7

2. Objectives Calculate the dot product of two vectors.
Calculate the angle between two vectors.
Use the dot product to determine if two vectors are orthogonal, parallel, or neither.

3. Vocabulary dot product orthogonal parallel
a product of two vectors formed by summing the product of their vertical components and the product of their horizontal components which produces a scalar number (not a vector)
two vectors are orthogonal if their dot product is 0
two vectors are parallel if the angle between them is either 0° or 180°

4. Formulas Dot Product of and OR
The Dot Product (MAT 170)
Formulas
Dot Product of and OR
Angle between two vectors (θ is the smallest non-negative angle between the two vectors)
and

5. Compute the dot product of each pair of vectors:
Our first step in find each of these dot products is to put the vectors into standard rectangular coordinates. The vector i in rectangular coordinates is i = 1i + 0j. The vector j in rectangular coordinates is j = 0i + 1j. is is is
continued on next slide

6. Compute the dot product of each pair of vectors:is
For this problem, the dot product formula to use is
In our problem a1 is 1, a2 is 1, b1 is 0, and b2 is 0.
continued on next slide

7. Compute the dot product of each pair of vectors:is
For this problem, the dot product formula to use is
In our problem a1 is 1, a2 is 0, b1 is 0, and b2 is 1.
continued on next slide

8. Compute the dot product of each pair of vectors:is
For this problem, the dot product formula to use is
In our problem a1 is 0, a2 is 0, b1 is 1, and b2 is 1.

9. Given the vectors u = 8i + 8j and v = —10i + 11j find the following.is
For this problem, the dot product formula to use is
In our problem a1 is 8, a2 is -10, b1 is 8, and b2 is 11.
continued on next slide

10. Given the vectors u = 8i + 8j and v = —10i + 11j find the following.is
For this problem, the dot product formula to use is
In our problem a1 is -10, a2 is 8, b1 is 11, and b2 is 8.
You should notice that when we computed the dot product with the vectors u•v first we got the same answer as when we switched the order of the vectors and calculated the dot product as v•u. This means that the dot product is commutative.
continued on next slide

11. Given the vectors u = 8i + 8j and v = —10i + 11j find the following.is
For this problem, the dot product formula to use is
In our problem a1 is -10, a2 is -10, b1 is 11, and b2 is 11.

12. Given the vectors u = 8i + 8j, v = —10i + 11j, and w = 9i + 7j find the following.
We start by doing the part in the parentheses
For this problem, the dot product formula to use is
continued on next slide

13. Given the vectors u = 8i + 8j, v = —10i + 11j, and w = 9i + 7j find the following.
We will now do the dot product of our result from the previous slide and the vector u.
For this problem, the dot product formula to use is
In our problem a1 is 8, a2 is -1, b1 is 8, and b2 is 18.
continued on next slide

14. Given the vectors u = 8i + 8j, v = —10i + 11j, and w = 9i + 7j find the following.
We start with doing the dot product of u and v and the dot product of u and w.
For this problem, the dot product formula to use is
continued on next slide

15. Given the vectors u = 8i + 8j, v = —10i + 11j, and w = 9i + 7j find the following.
Now we will add the dot products from the previous slide.
Notice that this is the same as the answer that we got for u•(v + w). This tells us that the dot product can be distributed over addition and subtraction.
continued on next slide

16. Given the vectors u = 8i + 8j, v = —10i + 11j, and w = 9i + 7j find the following.
We start with doing the dot product of v and u and the dot product of w and w.
For this problem, the dot product formula to use is
continued on next slide

17. Given the vectors u = 8i + 8j, v = —10i + 11j, and w = 9i + 7j find the following.
Now we multiply by the dot products we found by 3 and 12, respectively, and subtract the two results.

18. Find the angle θ in degrees measured between the vectors u = 10i + 3j and v = 1i — 7j .
For this problem, we want to use the formula for the angle between two vectors.
For this we need the to find the magnitude of each vector and the dot product of the two vectors.
For this problem, the dot product formula to use is
In our problem a1 is 10, a2 is 1, b1 is 3, and b2 is -7.
continued on next slide

19. Find the angle θ in degrees measured between the vectors u = 10i + 3j and v = 1i — 7j .
Now that we have the dot product, we need to find the magnitude of each vector. We can use the alternate formula for magnitude.
alternate magnitude formula.
Magnitude of u
Magnitude of v
continued on next slide

20. Find the angle θ in degrees measured between the vectors u = 10i + 3j and v = 1i — 7j .
Now we are set to plug everything into the formula to find the angle between the vectors u and v.

21. Determine if the pair of vectors is orthogonal, parallel, or neither.
The first step necessary to answer this question is to find the dot product of the two vectors. If the dot product is 0, then the vectors are orthogonal and we can stop. If the dot product is not 0, then we must go on to find the angle between the vectors.
For this problem, the dot product formula to use is
In our problem a1 is -8, a2 is 2, b1 is 3, and b2 is 16/3.
Since the dot product is 0, we can say that the vectors u and v are orthogonal.
continued on next slide

22. Determine if the pair of vectors is orthogonal, parallel, or neither.
For this problem, the dot product formula to use is
In our problem a1 is 7, a2 is 21, b1 is -3, and b2 is -9.
Since the dot product is not equal to 0, we know that the two vectors are not orthogonal. This means we need to continue with the formula for finding the angle between the two vectors by calculating the magnitude of each vector.
continued on next slide

23. Determine if the pair of vectors is orthogonal, parallel, or neither.
Now that we have the dot product, we need to find the magnitude of each vector. We can use the alternate formula for magnitude.
alternate magnitude formula.
Magnitude of u
Magnitude of v
continued on next slide

24. Determine if the pair of vectors is orthogonal, parallel, or neither.
Now we are set to plug everything into the formula to find the angle between the vectors u and v.
Since the angle between the two vectors is 0 degrees, the vectors are parallel.