Welcome to Engineering Mathematics 2

1. Welcome to Engineering Mathematics 2 We will cover 4 topics today 1. The Dot Product 2. The Cross Product 3. The Scalar Triple Product 4. The Vector Moment

2. Lecture Recap Last week we learnt that the determinant of a generic square matrix of order N is found using minors of the order MN-1. We also took linear simultaneous equations and solved them using determinants. Mathematically, an expansion about a column is expressed as We saw that this method could be used for any number of linear simultaneous equations. Finally we looked at basic vector algebra. We will expand on this today. An expansion about a row is expressed as

3. The Dot Product The multiplication of 2 vectors is defined in 2 different ways. The dot product is written as The first way is known as the dot (or scalar) product. Unsurprisingly the calculated result is a scalar. The cross product is written as The second way is known as the cross (or vector) product. The calculated result is a vector.

4. The Dot Product Question Find Suppose that two vectors are given in the following component form When The dot product is calculated as Therefore

5. The Dot Product Properties a) Commutative Property b) Distributive Property c) Connection with the magnitude (For two dimensions, omit the third component).

6. The Dot Product Question Prove that Using the properties on the previous slide we can determine that

7. The Dot Product z Consider the points P, A & B. B b a-b y θ P The cosine rule says that a A or x

8. The Dot Product Also, z B b a-b Comparing with the previous result of y θ P a A We can easily see that Where 0 ≤ θ ≥ 180. I.e. θ is the internal angle x

9. The Dot Product Question Given three points A : (1, 1, 1); B : (3, 2, 3) and C : (0, -1, 1). Find the angle between and . Let

10. The Dot Product In terms of i, j and k write down three dot products that are equal to 1. In the previous lecture we defined a unit vector. Three important unit vectors are In terms of i, j and k write down three dot products that are equal to 0.

11. The Cross Product Consider the following vectors. b b a a The vector a and b are both in the same plane whereas is perpendicular to both of them. i.e. If a right handed screw is turned from a to b then the screw will move in the direction of the unit normal. Hence, we can tell the direction of the normal using the “right hand rule”.

12. The Cross Product The cross product is defined in the following way The cross product has applications in problems that involve rotation (i.e. moments and angular velocities). The cross product is denoted by a bold multiplication sign or by a caret sign. I.e. Where; i, j & k are the unit vectors in the x, y & z directions respectively. ai & bi are the components of the a and b vectors in the 1, 2 & 3 directions (x, y, & z respectively). Or

13. The Cross Product Question Find the cross products a x b and b x a a = 2i – j + 3k; b = -i + 2j + 4k

14. The Cross Product Question Find the cross products a x b and b x a a = 2i – j + 3k; b = -i + 2j + 4k

15. The Cross Product Properties a) Commutative Property (the cross product does not commute) b) Distributive Property c) Scalar Multiplication d) If a and b are Parallel then

16. The Cross Product Calculate Consider the following cross product

17. The Cross Product We can deduce that Consider the following diagram z y b1 b2 b Therefore θ a = a1 Where is the unit normal to a and b. Hence, we can also write x

18. The Cross Product Question Let and be two vectors (from Q) representing the sides of a parallelogram. Show that the area of a parallelogram is equal to C Area = base x perpendicular height Area = B b Therefore, what is the area of a triangle in terms of a and b? A θ a Q

19. The Cross Product Question Let and be two vectors (from Q) representing the sides of a parallelogram. Show that the area of a parallelogram is equal to Area = base x perpendicular height C Area = B Therefore, what is the area of a triangle in terms of a and b? b A θ a Area = Q

20. The Scalar Triple Product If any two vectors are equal or parallel then The scalar quantity a.(b x c) is called a Scalar Triple Product. The triple product is calculated as If the lengths of the sides of a parallelepiped are given by the vectors a, b & c; then, Commutative Property If three vectors are coplanar when draw from the same point, then

21. The Scalar Triple Product Question Show that the points A : (1, 2, 2); B : (3, 4, 5); C : (-1, 0 , -1) lie on a plane through the origin. If the origin is designated by O; then, To show that a, b & c are coplanar we evaluate Therefore a, b & c are coplanar and pass through the origin.

22. The Scalar Triple Product Question What is the volume of the following parallelepiped? The parallelepiped is comprised of the following three vectors The volume is calculated using the triple product, i.e.

23. Applications Q The ‘Vector Moment’ F R Suppose that (in three dimensions) a force (F) is acting at a point (P) in a body. The moment or torque about an arbitrary point (Q) is defined as d θ P It can be seen that Thus, Hence, we can deduce that Where ‘d’ is the perpendicular distance between F and Q. This is shown in the following diagram. or Notice that R comes first and M is bold in the second equation.

24. Applications Question A force F = i- j + 2k acts at (1, 2, 1). Find its vector moment about the point (2, 1, 1). A force F = F1.i +F2.j acts at a point (a, b, 0). Find its vector moment about the origin. The moment is given by

25. Conclusion Essential reading for next week HELM Workbook 9.1 Basic Concepts of Vectors HELM Workbook 9.2 Cartesian Components of Vectors HELM Workbook 9.3 The Scalar Product HELM Workbook 9.4 The Vector Product HELM Workbook 9.5 Lines and Planes We have covered 4 topics today 1. The Dot Product 2. The Cross Product 3. The Scalar Triple Product 4. The Vector Moment