1. Have the following out on your table to show me Exercise A page 83 All 6 questions Blue worksheet of past parametric exam qu’s (3qu) If you don’t have with you any of the above you need to fill in a post it note with: Your full name Your tutor group List work you haven’t got with you and reason Can you differentiate ?

2. Aims: To know the Scalar Product formula and be able to use it to determine if two vector are perpendicular or parallel. To be able to use the scalar product to find the angle between vectors and direction vectors when a line is given in vector form. Vectors Lesson 5

3. The scalar product The result of this multiplication is a s_________/number quantity, hence the name. The scalar product of two vectors a and b is defined as: where θ is the angle between a and b when they are placed tail to tail. Note that θ is always taken to be between 0° and 180°. Two vectors can be multiplied together to give the scalar product, also known as the dot product. In general, if a = a 1 i + a 2 j + a 3 k and b = b 1 i + b 2 j + b 3 k then a.b = a 1 b 1 + a 2 b 2 + a 3 b 3. In general, if a = a 1 i + a 2 j + a 3 k and b = b 1 i + b 2 j + b 3 k then a.b = a 1 b 1 + a 2 b 2 + a 3 b 3.

4. Perpendicular and parallel vectors If a.b = 0 then a and b are perpendicular, as long as neither are the zero vector. In particular, for the unit base vectors i, j and k: We have seen that, since cos 90° = ______: If two vectors a and b are parallel, a.b = |a||b|. i.i =j.j =k.k =1×1 = 1 Also, if two vectors are parallel the angle between them is taken as 0°. Since cos 0° = ______ we can conclude from the scalar product that: In particular, for the unit base vectors i, j and k: i.j =i.k =j.i =j.k =k.i =k.j =

5. Scalar products of vectors in component form a.b = Find the scalar product of 10 a.b = Since a and b are both non-zero vectors, and their scalar product is _____, they must be p______________________. Prove that the vectors a = –3i + j + 2k and b = 2i + 8j – k are perpendicular.

6. On w/b a.b = Find the scalar product of 10 a.b = Find the scalar product of a = 4i +3j + 2k and b = 4i - 2j – 5k and state whether or not they are perpendicular.

7. Finding the angle between two vectors To do this we can write the scalar product in the form: For example, A useful application of the scalar product is in finding the acute angle between two vectors. Find the acute angle between the vectors a and b where and

8. On w/b Find the acute angle between the vectors a and b where and

9. 1.The position vectors of three points A, B and C relative to an origin O are given respectively by = 7i + 3j – 3k, = 4i + 2j – 4k = 5i + 4j – 5k. (i)Find the angle between AB and AC. [6] (ii)Find the area of triangle ABC. [2] Exam Questions

10. Exam Question (ii)

11. The position vectors of three points A, B and C relative to an origin O are given respectively by = 3i + 2j – 3k, = 2i + j – 4k = i + 4j – k. Find the angle between AB and AC. On w/bs

12. 2.Lines L 1, L 2 and L 3 have vector equations L 1 : r = (5i – j – 2k) + s(–6i + 8j – 2k), L 2 : r = (3i – 8j) + t( i + 3j + 2k), L 3 : r = (2i + j + 3k) + u(3i + cj + k). (i)Calculate the acute angle between L 1 and L 2. [4] (ii)Given that L 1 and L 3 are parallel, find the value of c. [2] Finding the angle between two vectors lines To find the angle between two vector lines we only need consider the direction of the lines. Re-call: r = a + t(b – a) Let p = and q =

13. Finding the angle between two vectors lines So the angle between them will be (ii) If the lines are parallel then they must have the same direction. Do exercise 9G page 127. Qu 2,4,6 and exercise 9H page 131. Qu’s 3, 4 & 5.

14. Lines L 1 and L 2 have vector equations L 1 : r = (i – j – 9k) + s(–6i + 8j – 4k), L 2 : r = (2i – 8j) + t( 3i + 2j + 1k), Calculate the acute angle between L 1 and L 2. On w/b Do exercise 9G page 127. Qu 2,4,6 and exercise 9H page 131. Qu’s 3, 4 & 5. Also Moodle h/w parametric equations. For test next Monday. –Can you differentiate sin 4 x?