Basic Math

1. Basic Math Vectors and Scalars Addition/Subtraction of Vectors Unit Vectors Dot Product

2. Scalars and Vectors (1) • Scalar – physical quantity that is specified in terms of a single real number, or magnitude • Ex. Length, temperature, mass, speed • Vector – physical quantity that is specified by both magnitude and direction • Ex. Force, velocity, displacement, acceleration • We represent vectors graphically or quantitatively: • Graphically: through arrows with the orientation representing the direction and length representing the magnitude • Quantitatively: A vector r in the Cartesian plane is an ordered pair of real numbers that has the form <a, b>. We write r=<a, b> where a and b are the components of vector v. Note: Both and r represent vectors, and will be used interchangeably.

3. Scalars and Vectors (2) • The components a and b are both scalar quantities. • The position vector, or directed line segment from the origin to point P(a,b), is r=<a, b>. • The magnitude of a vector (length) is found by using the Pythagorean theorem: • Note: When finding the magnitude of a vector fixed in space, use the distance formula.

4. Operations with Vectors (1) • Vector Addition/Subtraction The sum of two vectors, u=<u1, u2> and v=<v1, v2> is the vector u+v =<(u1+v1), (u2+v2)>. • Ex. If u=<4, 3> and v=<-5, 2>, then u+v=<-1, 5> • Similarly, u-v=<4-(-5), 3-2>=<9, 1>

5. Operations with Vectors (2) • Multiplication of a Vector by Scalar If u=<u1, u2> and c is a real number, the scalar multiple cu is the vector cu=<cu1, cu2>. • Ex. If u=<4, 3> and c=2, then cu=<(2·4), (2·3)> cu=<8, 6>

6. Unit Vectors (1) • A unit vector is a vector of length 1. • They are used to specify a direction. • By convention, we usually use i, j and k to represent the unit vectors in the x, y and z directions, respectively (in 3 dimensions). • i=<1, 0, 0> points along the positive x-axis • j=<0, 1, 0> points along the positive y-axis • k=<0, 0, 1> points along the positive z-axis • Unit vectors for various coordinate systems: • Cartesian: i, j, and k • Cartesian: we may choose a different set of unit vectors, e.g. we can rotate i, j, and k

7. Unit Vectors (2) • To find a unit vector, u, in an arbitrary direction, for example, in the direction of vector a, where a=<a1, a2>, divide the vector by its magnitude (this process is called normalization). • Ex. If a=<3, -4>, then <3/5, -4/5> is a unit vector in the same direction as a.

8. Dot Product (1) • The dot product of two vectors is the sum of the products of their corresponding components. If a=<a1, a2> and b=<b1, b2>, then a·b= a1b1+a2b2 . • Ex. If a=<1,4> and b=<3,8>, then a·b=3+32=35 • If θ is the angle between vectors a and b, then Note: these are just two ways of expressing the dot product • Note that the dot product of two vectors produces a scalar. Therefore it is sometimes called a scalar product.

9. Dot Product (2) • Convince yourself of the following: • Conclusion: After you define the direction of an arbitrary vector in terms of the Cartesian system, you can find the projection of a different vector onto the arbitrary direction. By dividing the above equation by the magnitude of b, you can find the projection of a in the b direction (and vice versa).