Vectors and Scalars

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HL Vectors Notes

1.

Vector or Scalar

Many physical quantities such as area, length, mass and temperature are completely described once the magnitude of the quantity is given. Such quantities are called “scalars.” Other quantities possess the properties of magnitude and direction. A quantity of this kind is called a “vector” quantity. Winds are usually described by giving their speed and direction; say 20 km/h north east. The wind speed and wind direction together form a vector quantity called the wind velocity. A force, for example, is characterized by its magnitude and direction of action. The force would not be completely specified by one of these properties without the other. The velocity of a moving body is determined by its speed and direction of motion. Acceleration and displacement are other examples of vector quantities.

A scalar is a quantity that has magnitude or size but no direction.

A vector is a quantity that has both magnitude and direction.

Displacement Vectors and Notation

[pic]

Vectors can be represented geometrically by arrows in 2- or 3-space; the direction of the arrow specifies the direction of the vector and the length of the arrow describes its magnitude. The first point in the arrow is called the initial point of the vector and the tip is called the terminal point. We shall denote vectors in lower case boldface type such as [pic] when using one letter to name the vector, and will use [pic] to denote the vector from A to B.

Note that the vector [pic] will reflect the displacement from A to B and not the distance traveled from A to B.

[pic]

Note that [pic]. They have the same length but their direction is different. However [pic]. [pic] is considered the negative of [pic].

Two vectors [pic] and [pic] are equal (equivalent) if they have the same length (magnitude) and the same direction and we write [pic]=[pic] . Geometrically, two vectors are equal if they are translations of one another; thus, the three vectors in the picture below are equal, even though they are in different positions.

[pic]

Example
[pic]

Length of a Vector
The length of a vector [pic] is also known as its modulus and it is written as [pic]. [pic]
[pic]
[pic]

Examples

[pic]

Position Vector

[pic]
Free Vector

[pic]

[pic]
[pic]

Addition of Vectors

[pic]

Two geometric vectors are added by the triangle law of addition ie. head-to-tail.

[pic]

[pic]

The addition of geometric vectors is commutative ie. p + q ’ q+ p.

Addition of geometric vectors is associative ie. (p + q) + r ’ p + (q+ r) .

[pic]

The triangle law of addition can be extended so that any number of vectors can be added by the head-to tail method.

[pic]
Example
[pic]
Studies Questions – Vectors

1.
[pic]
2.
[pic]

3.
[pic]
4.
[pic]
5.
[pic]

6.
[pic]
7.
[pic]

8.
[pic]
9.
[pic]

Vector Representation
[pic][pic][pic]
In general if [pic], then the magnitude [pic]

Vectors in 3 Dimensions

[pic]

Zero Vector

Because vectors are not affected by translation, the initial point of a vector [pic] can be moved to any convenient position by making an appropriate translation. If the initial and terminal points of a vector coincide, then the vector has length zero; we call this the zero vector and denote it by 0. The zero vector does not have a specific direction, so we will agree that it can be assigned any convenient direction in a specific problem.

[pic]

Multiplying a Vector by a Scalar

[pic]

Subtraction of Vectors

A vector is subtracted by adding its negative.
[pic]

Matrix Arithmetic and Vectors

[pic]

Unit Vectors

[pic]

5.2

Vector Multiplication – Scalar Product

There are two different kinds of vector multiplication: the scalar product (v.w) and the vector product (v x w).

The scalar...
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