PHYS 218 sec Review Chap. 1

Caution This presentation is to help you understand the contents of the textbook. Do not rely on this review for preparing exams. Solving and understanding all exercises and examples are very important. Also, try to solve the end of chapter problems.

Chap. 1 Unit: International System (SI unit) –Length: meter (m) –Mass: kilogram (kg) –Time: second (s) Unit prefixes See pps file for recitation (Sep. 1)

Vector Scalar: a quantity which can be described by a single number Vector: a quantity which has a magnitude and a direction –Mathematically, this is not the correct definition, but is enough for our purpose. –Notation: –Magnitude of a vector: –Be careful with the notations when you read other books or articles. A scalar can be multiplied to a vector –Changes the magnitude and the direction of the vector –E.g. When two vectors have the same magnitude and the same direction, they are identical.

Vector Addition To add two vectors in a graphical way, place the tail of the second vector at the head of the first vector, i.e., the starting point of the first vector becomes the starting point of the sum and the ending point of the second vector becomes the ending point of the sum. Vector Subtraction Subtracting vector B from vector A is identical to add vector A and vector (-B).

Dimension of space 1-dimension, one number is enough to specify the position. 0 x x y 2-dimension, two numbers (x,y) are needed to specify the position. x z y 3-dimension, three numbers (x,y,z) are needed to specify the position.

Unit vector A vector which has a magnitude of 1. It only specify the direction. Coordinate system Cartesian (or rectangular) coordinate system 2-dim. 3-dim. 1

Components of vectors You can write any vector as a linear combination of basis vectors. Polar coordinate system You can also use the magnitude (A) and the direction (  ) of a vector for writing the vector.

Relations between (A x,A y ) and (A,  )  is measured from the +x axis, rotating toward +y-axis This equation gives two solutions for . Draw diagrams to see which is the correct answer. In 3-dimensions In polar coordinates, we have (A, ,  ). But we do not discuss it.

Products of vectors Scalar (dot, inner) product This gives a scalar quantity.  

Scalar product using components Unit vectors: have magnitude 1 and perpendicular to each other This is a general expression. When the vectors are written or given with their components, you can always use this relation. You can use it as the definition of scalar product. But remember that this relation holds only when the basis vectors (i, j, k) are normal to each other.

Vector (cross, outer) product The vector product of two vectors gives another vector. The vector product is NOT commutative.

Vector product using components We use the right-handed coordinate system. The vector product is defined for 3-dim. vectors. If they are in cyclic order, then you have (+) sign. If not, you have (-) sign. Cyclic order (x y z)  (y z x)  (z x y) or (i j k) or (1 2 3) x yz

If you know how to calculate the determinant of a 3x3 matrix This is a coincidence. Matrix has no relation with the vector product. But if you are familiar with the matrix determinant, this is a good way to memorize the components of a vector product.