1.1 – 1.2 The Geometry and Algebra of Vectors

1. 1.1 – 1.2 The Geometry and Algebra of Vectors

2. Vectors in the Plane • Quantities that have magnitude but not direction are called scalars. • Ex: Area, volume, temperature, time, etc. • Quantities such as force, acceleration or velocity that have direction as well as magnitude are represented by directed line segments, called vectors. terminal point (head) B • The length of the vector is called • the magnitude and is denoted by A Initial point (tail)

3. Vectors are equivalent if they have the same length and direction (same slope). y • A vector is in standard positionif the initial point is at the origin. • The component formof this vector is: x (c,d) • If P are initial and terminal points of a vector, (a,b) Q x then the component form of v is: (a-c, b-d)

4. Example The component form of (-3,4) P is: (-5,2) Q The magnitude is v (-2,-2)

5. The magnitudeof is: then v is a zero vector: If then v is a unit vector. If and are called the standard unit vectors. e2 e1

6. Vector Operations Vector sum: Scalar Multiplication: Negative (opposite): Vector difference

7. The Parallelogram Rule u v (The Head-to-Tail Rule) u+v u + v is the resultant vector. v u u v u-v u - v is the resultant vector. v u

8. Linear Combination

9. Coordinates in Space • A three-dimensional coordinate systemconsists of: • 3 axes: x-axis, y-axis and z-axis • 3 coordinate planes: xy-plane, xz-plane and yz-plane • 8 octants. Each point is represented by an ordered triple Each vector is represented by z y x

10. Vectors in Space The magnitudeof is: then v is a zero vector : If then v is a unit vector. If and are called the standard unit vectors.

11. Vectors Operations Vector sum: Scalar Multiplication: Negative (opposite): Vector difference Vector v is parallel to u if and only if v = kufor some k.

12. Vectors in Rn and Zn • The set of all vectors with n real-valued components is denoted by Rn . Thus, a vector in Rnhas the form • R2 is the set of all vectors in the plane. • R3 is the set of all vectors in three-dimensional space. • Z3 is the set of all vectors in three-dimensional space whose components are integers. • Z3is the set {0, 1, 2} with special operations (Integer modulo 3)

13. Algebraic Properties of Vectors

14. Definitions Thedot product of u and vis defined by (Read “u dot v”) The dot product is also called scalar product. • Two vectors u and v are orthogonal • if they meet at a right angle. • if and only if u ∙ v = 0. Thedistance between vectors u and v is defined by

15. Example

16. Properties Another form of the Dot Product:

17. Example Find the angle between vectors u and v:

18. Vector Components u w2 w1 v • Let u and v be nonzero vectors. • w1 is called the vector componentof ualongv • (or projection of u onto v), and is denoted by projvu • w2 is called the vector component of uorthogonal tov

19. Examples Write u as a linear combination of standard unit vectors. Find u + v and 2u – 3v. Are u and v parallel? orthogonal? Find the angle between u and v. Find the magnitude of v. Normalize vector v. Find the projection of u onto v. Find the vector component of u orthogonal to v. Find the projection of v onto u. Find the distance between u and v.