1. DOT PRODUCT CROSS PRODUCT APPLICATIONS
VECTORS
DOT PRODUCT
CROSS PRODUCT
APPLICATIONS
NHAA/IMK/UNIMAP

2. DIRECTIONS OF ANGLES & DIRECTIONS OF COSINES
- Are the angles that the vector OP makes with positive axis
- Knows as the direction angles of vector OP
DIRECTION OF COSINES
NHAA/IMK/UNIMAP

3. Find the direction cosines and direction angles of:
Example 1
Find the direction cosines and direction angles of:
NHAA/IMK/UNIMAP

4. DOT PRODUCT Also known as inner product or scalar product
The result is a scalar
If and then:
NHAA/IMK/UNIMAP

5. Example 2
If and :
NHAA/IMK/UNIMAP

6. DOT PRODUCT Angle Between 2 Vectors
If the vectors lies on the same line or parallel to each other, then
NHAA/IMK/UNIMAP

7. Find the angles between and
Example 3
Find the angles between and
NHAA/IMK/UNIMAP

8. DOT PRODUCT
Properties of Dot Product
NHAA/IMK/UNIMAP

9. CROSS PRODUCT The result is a vector
If and then: (determinant of the matrix)
NHAA/IMK/UNIMAP

10. Find the cross product between and
Example 3
Find the cross product between and
NHAA/IMK/UNIMAP

11. CROSS PRODUCT Properties of Cross Product if u and v are parallel
NHAA/IMK/UNIMAP

12. APPLICATIONS PROJECTIONS
The vector projection of u = onto a nonzero vector
Is the vector determined by dropping a perpendicular from Q to the line PS.
NHAA/IMK/UNIMAP

13. APPLICATIONS PROJECTIONS Scalar projection Vector projection
The magnitude of the vector projection:
Vector projection
NHAA/IMK/UNIMAP

14. APPLICATIONS AREA OF TRIANGLE & PARALLELOGRAM
The magnitude of uxv is the area of parallelogram
NHAA/IMK/UNIMAP

15. APPLICATIONS AREA OF TRIANGLE Half of the area of parallelogram v u
NHAA/IMK/UNIMAP

16. Parametric Equations of a Line in Equations of Planes
APPLICATIONS
Parametric Equations of a Line in
Equations of Planes
Distance from a Point to the Plane
NHAA/IMK/UNIMAP

17. Line L is the set of all points P(x,y,z) for which parallel to :
APPLICATIONS
Lines & Line Segment in Space
Parametric Equations z y x
P0(x0,y0,z0) L
P(x,y,z) v
Line L is the set of all points P(x,y,z) for which parallel to :
NHAA/IMK/UNIMAP

18. APPLICATIONS Therefore the parametric equation for L : Parametric
NHAA/IMK/UNIMAP

19. Cartesian equation:
NHAA/IMK/UNIMAP

20. Example 3: Find parametric and Cartesian equation for the line passes through Q(-2,0,4) and parallel to
NHAA/IMK/UNIMAP

21. Example 4: Find the parametric equation for the line passes through P(-3,2,-3) and Q(1,-1,4)
NHAA/IMK/UNIMAP

22. APPLICATIONS Distance from point S to line Lv u L
From the properties of Cross Product
Formula of Distance from point S to L
NHAA/IMK/UNIMAP

23. Example 5: Find the distance from the point S(1,1,5) to the line
NHAA/IMK/UNIMAP

24. APPLICATIONS Lines of Intersection n1 Finding v : n2 v
Line of intersection
NHAA/IMK/UNIMAP

25. Example 8: Find a vector parallel to the line of intersection of the planes
NHAA/IMK/UNIMAP

26. Example 9: Find the parametric equation for the line in which the planes Intersect.
NHAA/IMK/UNIMAP

27. Example 9: Find the point where the line Intersects the plane
NHAA/IMK/UNIMAP

28. APPLICATIONS Equation of Planes
P0(x0,y0,z0)
P(x,y,z)
Vector is on the plane M and vector which is perpendicular to M known as normal vector, n
From the properties of Dot Product
NHAA/IMK/UNIMAP

29. APPLICATIONS
Equation of Planes
Normal vector n :
NHAA/IMK/UNIMAP

30. APPLICATIONS
Let and
EQUATION OF PLANE
NHAA/IMK/UNIMAP

31. Example 6: Find an equation of plane through P0(-3,0,7) perpendicular to
NHAA/IMK/UNIMAP

32. Example 7: Find equation of plane through 3 points:
NHAA/IMK/UNIMAP

33. APPLICATIONS Distance from a Point to the Plane P n D P0
NHAA/IMK/UNIMAP

34. APPLICATIONS
Equation of plane,
With n = <a,b,c>
NHAA/IMK/UNIMAP

35. Example 10: Find the distance from S(1,1,3) to the plane
NHAA/IMK/UNIMAP