Physics 141Mechanics Lecture 3 Vectors Motion in 2-dimensions or 3-dimensions has to be described by vectors. In mechanics we need to distinguish two types of physical quantities: scalars or vectors. A scalar has only magnitude, large or small, positive or negative, such as time, mass, temperature, etc. A single value in the right unit, together with a sign, completely specify a scalar. A vector has not only the magnitude but also direction, such as displacement, velocity, acceleration, force, etc. A displacement cannot be specified by giving only the magnitude. The direction of the displacement must also be given.

Expression of a Vector Graphically, a vector is represented by a line with an arrow. The length of the line is the magnitude, and the direction of the line represent the vector’s direction. A vector is typically represented by a bold letter, say, a. In handwriting, typically one put an arrow on top of the letter to represent a vector. The magnitude of a =|a| = a. A displacement is a vector. Vectors (of the same kind) can be added. If you go north 1 mi and then east 1 mi, your displacement can be represented by one vector pointing to the northeast with magnitude of √2 mi. a

Addition of Vectors The vector sum c=a+b is a new vector whose magnitude and direction are determined by the same quantities of the two adding vectors. Vector additions are commutative, a+b=b+a Vector additions are associative, (a+b)+c=a+(b+c) a and -a are two different vectors of the same magnitude but opposite directions. a-b=a+(-b) c b a a -a

Multiplication by Scalars and Components A vector can be multiplied by a number or a scalar. ca is in the same direction of a, and the magnitude |ca|=c|a|. A vector of unit magnitude is called a unit vector. For any vector a, a/|a| is a unit vector along a, often written as To describe a vector quantitatively, we use a right-handed coordinate system with x, y, and z axes, and the unit vectors along the three axes are i, j, k, respectively. Suppose a point has coordinates (x,y,z), then the vector r=xi+yj+zk represents a position vector pointing from the origin to the point. The numbers (x,y,z) are called the components of r.

Components, Magnitude, and Angles Any vector a in 3-D can be expressed uniquely by specifying its components along x, y, and z. (1) The magnitude (2) The angles of a with the three axes are (3)

Example: Let’s look at 2-D case From the triangle it is clear that (4) Since (5) We can also write (6) x y a axax ayay xx yy

Adding Vectors by Components Expressing vectors by their components in the coordinate system makes it feasible to do vector-related calculations. One example is vector addition. All we have to do now is just to add the same component. c=a+b => (7) It’s usually much easier than using the graphic method.

Scalar Product The scalar product is one type of multiplication of two vectors, defined as ab=ab cos  (8) where  is the angle between the vectors a and b. The result is a scalar. Since it has a dot in it, it’s also called a dot product. Look at the cos  if b is along the x-axis, a cos  gives the x component of a. In fact, the scalar product is the product of the magnitude of one vector with the component of the other along the former. We will see how the concept of scalar product is useful when we study the relations among force, displacement, and work. The scalar product satisfies the following ab=ba, (a+b)c=ac+bc (9)

Scalar Product in Terms of Component Just as in the case of vector addition, the scalar product is more feasible in terms of components. First let’s notice that if  =90 degrees, cos  =0, and ab=0. This condition is certainly met for the unit vectors along the axes, i,j,k. So ij=0 and ii=1 (=cos 0). We then have (10) It can often be used to calculate the angle between two vectors (11) It can also be used to check if two vectors are perpendicular to each other.

Example: If a = 9i - 5k, b = 2i + 3j, what is the angle between them? Solution: ab = 9 x x 3 + (-5) x 0 = 18 Thus the angle between a and b is  = arccos(ab/ab)) = arccos(18/(10.30 x 3.61)) = 61.0 degrees

Vector Product The vector product is another type of vector multiplication whose result is a new vector, denoted as c = a x b (12) The magnitude of the vector product is c = ab sin  (13) The direction of the vector product is perpendicular to the plane formed by a and b, and it obeys the right hand rule: if you curl your right hand along the small angle from a to b, then your thumb will point to the direction of the cross product. If a and b are parallel to each other, then the vector product c = 0. BTW, the vector product is also called the cross product.

Features of Vector Product It is easy to verify by using your right hand that a x b = - b x a, and it’s obvious that a x a = 0. For the unit vectors i, j, k, there exist relations such as i x j = k, j x k = i, k x i = j, and i x i = 0, j x j = 0, k x k =0. If and The components of c = a x b are (14)