Vectors, Addition of Vectors, Triangle Law of Addition of Vectors, Addition of More Than Two Vectors, Parallelogram Law of Addition of Vectors, Negative of a Vector

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Addition of Vectors:

  • Recall that you have learnt four fundamental operations viz. addition, subtraction, multiplication and division on numbers.

  • The addition (subtraction) of vectors is different from that of numbers (scalars).

  • In fact, there is the concept of resultant of two vectors (these could be two velocities, two forces etc.) We illustrate this with the help of the following example:

  • Let us take the case of a boat-man trying to cross a river in a boat and reach a place directly in the line of start.

  • Even if he starts in a direction perpendicular to the bank, the water current carries him to a place different from the place he desired., which is an example of the effect of two velocities resulting in a third one called the resultant velocity.

  • Thus, two vectors with magnitudes 3 and 4 may not result, on addition, in a vector with magnitude.

  • It will depend on the direction of the two vectors i.e., on the angle between them.

  • The addition of vectors is done in accordance with the triangle law of addition of vectors.

Triangle Law of Addition of Vectors:

A vector whose effect is equal to the resultant (or combined) effect of two vectors is defined as the resultant or sum of these vectors. This is done by the triangle law of addition of vectors.

Triangle Law of Addition of Vectors

Triangle Law of Addition of Vectors

Triangle Law of Addition of Vectors

In the adjoining Fig. vector is the resultant or sum of vectors and is written as

i.e.

You may note that the terminal point of vector is the initial point of vector and the initial point of is the initial point of and its terminal point is the terminal point of .

Addition of More Than Two Vectors:

Addition of more than two vectors is shown in the adjoining figure

Addition of more than two Vectors

Addition of More Than Two Vectors

Addition of more than two Vectors

The vector is called the sum or the resultant vector of the given vectors.

Parallelogram Law of Addition of Vectors:

Recall that two vectors are equal when their magnitude and direction are the same. But they could be parallel [refer to Fig.].

Parallelogram Law of Addition of Vectors

Parallelogram Law of Addition of Vectors

Parallelogram Law of Addition of Vectors

See the parallelogram in the adjoining figure:

We have,

But

Which is the parallelogram law of addition of vectors. If two vectors are represented by the two adjacent sides of a parallelogram, then their resultant is represented by the diagonal through the common point of the adjacent sides.

Negative of a Vector:

For any vector , the negative of is represented by . The negative of is the same as .

Thus, and . It follows from definition that for any vector ,

The Difference of Two Given Vectors:

The Difference of Two Given Vectors

The Difference of Two Given Vectors

The Difference of Two Given Vectors

For two given vectors and , the difference is defined as the sum of and the negative of the vector . i.e., .

In the adjoining figure if then, in the parallelogram, and

Example:

Show by a diagram

Solution:

Complete the parallelogram OABC

Complete the Parallelogram OABC

Complete the parallelogram OABC

From the adjoining figure, resultant

….. (i)

Complete the parallelogram

….. (ii)

[From (i) and (ii)]

Developed by: