Vectors, Objectives, Scalars and Vectors, Vectors as a Directed Line Segment

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Objectives:

After studying this lesson, you will be able to:

  • Explain the need of mentioning direction;

  • Define a scalar and a vector;

  • Distinguish between scalar and vector;

  • Represent vectors as directed line segment;

  • Determine the magnitude and direction of a vector;

  • Classify different types of vectors-null and unit vectors;

  • Define equality of two vectors;

  • Define the position vector of a point;

  • Add and subtract vectors;

  • Multiply a given vector by a scalar;

  • State and use the properties of various operations on vectors;

  • Comprehend the three dimensional space;

  • Resolve a vector along two or three mutually perpendicular axes;

  • Derive and use section formula; and

  • Define scalar (dot) and vector (cross) product of two vectors.

  • Define and understand direction cosines and direction ratios of a vector.

  • Define triple product of vectors.

  • Understand scalar triple product of vectors and apply it to find volume of a rectangular parallelepiped.

  • Understand coplanarity of four points.

Scalars and Vectors:

  • A physical quantity which can be represented by a number only is known as a scalar i.e., quantities which have only magnitude.

  • Time, mass, length, speed, temperature, volume, quantity of heat, work done etc. are all scalars.

  • The physical quantities which have magnitude as well as direction are known as vectors.

  • Displacement, velocity, acceleration, force, weight etc. are all examples of vectors.

Vectors as a Directed Line Segment:

  • You may recall that a line segment is a portion of a given line with two end points. Take any line (called a support).

  • The portion of with end points and is called a line segment.

Vectors as a Directed Line Segment

Vectors as a Directed Line Segment

Vectors as a Directed Line Segment

  • The line segment along with direction from to is written as and is called a directed line segment.

  • and are respectively called the initial point and terminal point of the vector .

  • The length is called the magnitude or modulus of and is denoted by.

  • In other words the length.

  • Scalars are usually represented by etc. whereas vectors are usually denoted by etc. Magnitude of a vector i.e., is usually denoted by ‘’.

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