Physics ADDITION AND SUBTRACTION OF VECTORS - GRAPHICAL METHOD
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### Topics Covered

• Multiplication of Vectors by Real Numbers
• Addition & Subtraction of Vectors - Graphical Method
• Triangle Method
• Parallelogram Method

### Multiplication of Vectors by Real Numbers

text(Multiplication with a Positive Number)
Multiplying a vector vecA with a positive number lamda gives a vector whose magnitude is changed by the factor lamda but the direction is the same as that of vecA.
|lamdavecA|=lamda|vecA| if lamda > 0

For example, if vecA is multiplied by 2, the resultant vector 2vecA is in the same direction as vecA and has a magnitude twice of |vecA| as shown in Fig. (a).

text(Multiplication with a Negative Number)
Multiplying a vector vecA by a negative number lamda gives a vector lamdavecA whose direction is opposite to the direction of vecA and whose magnitude is –lamda times |vecA|.

Multiplying a given vector vecA by negative numbers, say –1 and –1.5, gives vectors as shown in Fig (b).

### Addition & Subtraction of Vectors - Graphical Method

Graphical methods of vectors addition and subtraction -
• Triangle Method
• Parallelogram Method

### Head-to-Tail or Triangle Method of Vector Addition

Let us consider two vectors vecA and vecB that lie in a plane as shown in Fig. (a). The lengths of the line segments representing these vectors are proportional to the magnitude of the vectors.
To find the sum vecA + vecB, we place vector vecB so that its tail is at the head of the vector vecA, as in Fig. (b). Then, we join the tail of vecA to the head of vecB. This line OQ represents a vector vecR, that is, the sum of the vectors vecA and vecB. Since, in this procedure of vector addition, vectors are arranged head to tail, this graphical method is called the head-to-tail method.

The two vectors and their resultant form three sides of a triangle, so this method is also known as text(triangle method of vector addition).

• If we find the resultant of vecB + vecA as in Fig. (c), the same vector vecR is obtained. Thus, vector addition is commutative.
vecA + vecB = vecB + vecA
• The addition of vectors also obeys the associative law as illustrated in Fig. (d). The result of adding vectors vecA and vecB first and then adding vector vecC is the same as the result of adding vecB and vecC first and then adding vector vecA.
(vecA + vecB) + vecC = vecA + (vecB + vecC)

text(Null Vector or Zero Vector)
Consider two vectors vecA and –vecA. Their sum is vecA + (–vecA). Since the magnitudes of the two vectors are the same, but the directions are opposite, the resultant vector has zero magnitude and is represented by vec0 called a text(null vector) or a text(zero vector).
vecA – vecA = vec0
|vec0|= 0
Since the magnitude of a null vector is zero, its direction cannot be specified.

The null vector also results when we multiply a vector A by the number zero. The main properties of vec0 are -
vecA + vec0 = vecA
lamda vec0 = vec0
0 vecA = vec0

text(Subtraction of Vectors)
Subtraction of vectors can be defined in terms of addition of vectors. We define the difference of two vectors vecA and vecB as the sum of two vectors vecA and –vecB.
vecA – vecB = vecA + (–vecB)
It is shown in Fig (e,f). The vector –vecB is added to vector vecA to get vecR_2 = (vecA – vecB).

### Parallelogram Method

Suppose we have two vectors vecA and vecB. To add these vectors, we bring their tails to a common origin O as shown in Fig. (a).
Then we draw a line from the head of vecA parallel to vecB and another line from the head of vecB parallel to vecA to complete a parallelogram OQSP. Now we join the point of the intersection of these two lines to the origin O.

The resultant vector vecR is directed from the common origin O along the diagonal (OS) of the parallelogram [Fig. (b)].

In Fig. (c), the triangle law is used to obtain the resultant of vecA and vecB and we see that the two methods yield the same result. Thus, the two methods are equivalent.
Q 3015778669

Rain is falling vertically with a speed of 35 m s^(–1). Winds starts blowing after sometime with a speed of 12 m s^(–1) in east to west direction. In which direction should a boy waiting at a bus stop hold his umbrella ?

Solution:

The velocity of the rain and the wind are represented by the vectors v_r and v_w in Fig. and are in the direction specified by the problem. Using the rule of vector addition, we see that the resultant of v_r and v_w is R as shown in the figure. The magnitude of R is

R = sqrt(v_r^2+v_w^2) = sqrt((35)^2+(12)^2) m s^(-1) = 37 ms^(-1)

The direction θ that R makes with the vertical is given by tantheta = v_w/v_r = 12/35 = 0.343

or theta = tan^(-1) (0.343) = 19^0

Therefore, the boy should hold his umbrella in the vertical plane at an angle of about 19^0 with the vertical towards the east.