1. VECTORS

2. A vector is a quantity that has both magnitude and direction. It is represented by an arrow. The length of the vector represents the magnitude and the arrow indicates the direction of the vector. Two vectors are equal if they have the same direction and magnitude (length). Blue and orange vectors have same magnitude but different direction. Blue and green vectors have same direction but different magnitude. Blue and purple vectors have same magnitude and direction so they are equal.

3. Component Form of a Vector The component form of the vector with initial point P = (p 1, p 2 ) and terminal point Q = (q 1, q 2 ) is PQ The magnitude (or length) of v is given by

4. Find the component form and length of the vector v that has initial point (4,-7) and terminal point (-1,5) -22 4 6 4 2 -2 -4 -6 -8 Let P = (4, -7) = (p 1, p 2 ) and Q = (-1, 5) = (q 1, q 2 ). Then, the components of v = are given by v 1 = q 1 – p 1 = v 2 = q 2 – p 2 = -1 – 4 = -5 5 – (-7) = 12 Thus, v = and the length of v is

5. P Q Initial Point Terminal Point magnitude is the length direction is this angle How can we find the magnitude if we have the initial point and the terminal point? The distance formula How can we find the direction? (Is this all looking familiar for each application? You can make a right triangle and use trig to get the angle!)

6. Q Terminal Point direction is this angle Although it is possible to do this for any initial and terminal points, since vectors are equal as long as the direction and magnitude are the same, it is easiest to find a vector with initial point at the origin and terminal point (x, y). If we subtract the initial point from the terminal point, we will have an equivalent vector with initial point at the origin. P Initial Point A vector whose initial point is the origin is called a position vector

7. To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words). Initial point of v Move w over keeping the magnitude and direction the same. To add vectors, we put the initial point of the second vector on the terminal point of the first vector. The resultant vector has an initial point at the initial point of the first vector and a terminal point at the terminal point of the second vector (see below--better shown than put in words). Terminal point of w

8. The negative of a vector is just a vector going the opposite way. A number multiplied in front of a vector is called a scalar. It means to take the vector and add together that many times.

9. Using the vectors shown, find the following:

10. Vectors are denoted with bold letters (a, b) This is the notation for a position vector. This means the point (a, b) is the terminal point and the initial point is the origin. We use vectors that are only 1 unit long to build position vectors. i is a vector 1 unit long in the x direction and j is a vector 1 unit long in the y direction. (3, 2)

11. If we want to add vectors that are in the form ai + bj, we can just add the i components and then the j components. Let's look at this geometrically: When we want to know the magnitude of the vector (remember this is the length) we denote it Can you see from this picture how to find the length of v?

12. A unit vector is a vector with magnitude 1. If we want to find the unit vector having the same direction as a given vector, we find the magnitude of the vector and divide the vector by that value. If we want to find the unit vector having the same direction as w we need to divide w by 5. Let's check this to see if it really is 1 unit long.

13. If we know the magnitude and direction of the vector, let's see if we can express the vector in ai + bj form. As usual we can use the trig we know to find the length in the horizontal direction and in the vertical direction.

14. Acknowledgement I wish to thank Shawna Haider from Salt Lake Community College, Utah USA for her hard work in creating this PowerPoint. www.slcc.edu Shawna has kindly given permission for this resource to be downloaded from www.mathxtc.com and for it to be modified to suit the Western Australian Mathematics Curriculum.www.mathxtc.com Stephen Corcoran Head of Mathematics St Stephen’s School – Carramar www.ststephens.wa.edu.au

15. Vector Operations The two basic operations are scalar multiplication and vector addition. Geometrically, the product of a vector v and a scalar k is the vector that is times as long as v. If k is positive, then kv has the same direction as v, and if k is negative, then kv has the opposite direction of v. v ½ v2v -v

16. Definition of Vector Addition & Scalar Multiplication Let u = and v = be vectors and let k be a scalar (real number). Then the sum of u and v is u + v = and scalar multiplication of k times u is the vector

17. Vector Operations Ex. Let v = and w =. Find the following vectors. a. 2v b. w – v -22 6 10 8 4 2 -2 -4 v 2v 1 2 3 4 4 3 2 1 5 w -v w - v

18. Writing a Linear Combination of Unit Vectors Let u be the vector with initial point (2, -5) and terminal point (-1, 3). Write u as a linear combination of the standard unit vectors of i and j. -22 4 6 4 2 -2 -4 -6 -8 (2, -5) u (-1, 3) Solution -22 6 10 8 4 2 -2 -4 Graphically, it looks like… -3i 8j

19. Writing a Linear Combination of Unit Vectors Let u be the vector with initial point (2, -5) and terminal point (-1, 3).Write u as a linear combination of the standard unit vectors i and j. Begin by writing the component form of the vector u.

20. Unit Vectors u = unit vector Find a unit vector in the direction of v =

21. Vector Operations Let u = -3i + 8j and let v = 2i - j. Find 2u - 3v. 2u - 3v = 2(-3i + 8j) - 3(2i - j) = -6i + 16j - 6i + 3j = -12i + 19 j

22. Dot Products The Definition of the Dot Product of Two Vectors The dot product of u = and v = is Ex.’s Find each dot product.

23. Properties of the Dot Product Let u, v, and w be vectors in the plane or in space and let c be a scalar.

24. Let FindFirst, find u. v Find u. 2v= 2(u. v)= 2(-14) = -28

25. The Angle Between Two Vectors If is the angle between two nonzero vectors u and v, then Find the angle between

26. Definition of Orthogonal Vectors (90 degree angles) The vectors u and v are orthogonal if u. v = 0 Are the vectors orthogonal? Find the dot product of the two vectors. Because the dot product is 0, the two vectors are orthogonal. End of notes.