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Unit 38 Vectors

In this unit, we explore the concepts of equivalent vectors, opposite direction vectors, vector addition, and geometry. We will analyze pairs of vectors to determine equivalence and directionality, calculate vector components, and express relationships between points in terms of vector notation. Through various exercises including parallelogram properties and collinearity of points, we will build a deeper understanding of vector expressions and their applications in geometry. This unit is integral for mastering vector operations and geometric interpretations.

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Unit 38 Vectors

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  1. Unit 38Vectors

  2. Unit 38 38.1 Equal Vectors

  3. In the following diagram, state Which pairs of vectors are equivalent and which pairs of vectors are opposite directions Equivalent vectors: a and bb andi f and g Vectors in opposite directions c and b e and j ? ? ? ? ?

  4. Unit 38 38.2 Components

  5. Given that , , calculate ? ? ? ? ? ? ? ? ? ? Does ? No

  6. Unit 38 38.3 Vector Expressions

  7. Mark clearly on the diagram (a) the point P such that (b) the point Q such that Q P Write down each of the following in terms of d and/or e. (a) (b) (c) ? ? ? ? ? ?

  8. Unit 38 38.4 Addition and Subtraction of Vectors

  9. Given that On the grid below, illustrate the following vectors (i) b – c (ii) c + a (iii) a + b (iv) a + c – b (i) b - c - c b (iii) (ii) c + a b a c a + b (iv) a a + c - b a + c - b

  10. Unit 38 38.5 Vector Geometry 1

  11. Express, in terms of u and v, (a) (b) (c) , where M is the midpoint of AC, ? ? ?

  12. Express, in terms of u and v, (d) (e) , where N is the midpoint of BD, (f) ? ? ? ? ?

  13. Hence and What can you deduce about points M and N? They are coincident.

  14. Unit 38 38.6 Vector Geometry 2

  15. In the figure above, ABCD is a parallelogram such that and . The point P is on DB such that • Express in terms of xandy, • (i) (ii) (iii) • Solution (i) (ii) (iii) ? ? ? ?

  16. In the figure above, ABCD is a parallelogram such that and . The point P is on DB such that (b) Show that Solution ? ?

  17. In the figure above, ABCD is a parallelogram such that and . The point P is on DB such that • (c) Given that E is the midpoint of DC, prove that A, P and E are collinear. • Solution ? ? so from part (a) ? from part (b) ? Hence A, P and E are collinear ?

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