Vectors: Introduction (1)
INTRODUCTION TO VECTORS
You must have heard that vectors are quantities which have magnitude and direction. Right? But how to bring them into geometry? Suppose we have a force of 4 N applied to a mass in the horizontal direction . How do we represent that in Geometry? Let us consider a line segment of length 4 units. Will it suffice? No. We have to show the direction as well. How do we do that? Suppose our line segment is AB and is starting at the point A and ending in B. How to show the direction now?Direction can be associated with some kind of movement. Right? Notice that when we start to draw the line segment we start at A and move in a certain way. Why not move in the direction of the applied force, that is in the horizontal way to the right and end up in B after moving a distance of 4 units. To show the motion in the manner as is indicated let us put an arrowhead at the end point B. This will certainly make sense.
Construct the above vector in the applet below . Make sure it is of magnitude 4 and indicates a horizontal direction to the right. Then construct another vector in the vertically upward direction having magnitude 3 units.
EQUALITY OF VECTORS
Two vectors will be equal if and only if their magnitude and directions match. Right ? Suppose we have a vector and another vector . How can these two be equal?
Change the positions of A,B, C and D. Drag the points A,B,C,D so that the two vectors become equal.
A vector whose magnitude denoted by is equal to 1 is defined as a unit vector. Note that a unit vector may have any direction. There are two special unit vectors denoted as and . The first one has a direction along positive x-axis and the second one along positive y-axis.
Drag the point A.
Scalar multiple of a vector
Suppose we have a vector . Let be any scalar then their product is denoted as which has the magnitude and has the same direction as if and has opposite direction if .
Observe how the value of a becomes the scalar multiple.
This applet defines how to add two vectors.
Horizontal and vertical component of a vector.
Any vector in a plane can be expressed as where are two scalars and and are the two unit vectors defined above.
If and then what is and what is ?