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Lecture 2: Vectors 向量 (chapter 3 of Halliday)

Lecture 2: Vectors 向量 (chapter 3 of Halliday). A vector has both magnitude and direction, e.g. displacement is a vector. But some physical quantities are not vectors, they are scalars, e.g. temperature, mass, time, density, etc….

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Lecture 2: Vectors 向量 (chapter 3 of Halliday)

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  1. Lecture 2: Vectors 向量 (chapter 3 of Halliday) A vector has both magnitude and direction, e.g. displacement is a vector. But some physical quantities are not vectors, they are scalars, e.g. temperature, mass, time, density, etc… For displacement, both magnitude and direction is needed to fully specify the change of position, say from A to B. However, the path is not specified. We will learn: Vector addition, subtraction, vector products – scalar product and cross product.

  2. Sum of two vectors: Note: the order is not important.

  3. Vector subtraction: Note: d+b=a Question:

  4. Components of vectors A better way to do addition or subtraction is to resolve the vector into components: x and y components. i.e. (a, q) -> (ax, ay) (here we consider vectors on a plane, 2D case) To obtain magnitude and angle from the components, one can use:

  5. Note that the definition of q will give a negative by if q is larger than 180°, which means the vector is pointing downward. For what q,bx is negative?

  6. Unit vector A formal way to express a vector into components is to use unit vectors: where the unit vectors have magnitude of one unit. So essentially, unit vectors only represent the directions along x, y and z. Addition by components Subtraction is done by simply replacing the + sign by – sign.

  7. We can take dnet,x = 8.2cm instead, so the angle is 24.86°. The final angle is 180°-24.86°=155°.

  8. Multiplying vectors Dot product (or scalar product): If a is perpendicular to b, i.e. f=90°, the dot product=0. If a is parallel to b, i.e. f=0°, the dot product=ab. In components form: In physics, dot product is commonly used. For example, the work done W = F·s. (F is force vector, s is displacement vector)

  9. A simply method to find the angle between two vectors.

  10. Cross product (vector product) The product is a vector (a dot product is a scalar). The magnitude is given by: The direction of c is deduced by the right-hand rule: Note that i.e. the order of the product is important. In component form: Example of cross product in physics:

  11. Unlike dot product, CxD=0 if they are parallel, and CxD = CD when they are perpendicular.

  12. (1) + x; (2) + z; (3) + z • (1) – x; (2) – z; (3) – z Problems: 23, 24, 33

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