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.

Unit Vectors

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

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 ?

Select all that apply

A
=3 and =0
B
=-4 and =-3
C
=-4 and =3

Check my answer (3)