(i) Addition of vectors
Since vectors possess direction in addition to the magnitude as possessed by scalar quantities. So they can not be added by using law of algebra.Let us discuss the addition of two vectors by taking example of displacement vectors.
(a) When two vector act in the same direction :
Let a particle is displaced through a distance of 5m along west to east direction and let it be represented in magnitude and direction by p,let this particle be again displaced through 3m along same direction i.e.,west to east,let it be represented by 'q' then the resultant displacement of two vectors will be
→ →
P + Q = 8 m along west to east direction.
(b) When two vector are in opposite direction :
Let these two displacement vectors 'p'(vector) and 'q'(vector) be in opposite direction p (vector) is acting in west to two displacement vector will be 2m,but it will act in direction along which vector with greater magnitude acts i.e., along west to east direction.
(c) When two or more vectors act at some angle to each other : In this case,we make use of either (a) triangle law,(b) parallelogram law or (c) polygon law of vectors.Let us study them one by one.
(ii) Triangle law of vectors
If two vector can be represented both in magnitude and direction by the sides of a triangle taken in the same order,then the resultant is represented completely in magnitude and direction,by the third side of the triangle taken in the opposite order.
Suppose we have to add two vectors 'p' and 'q'.Place the initial point (tail) of 'q' on the terminal point (head) of p.Complete the triangle to get a new vector 'p'+'q' (vector) starting from tail of p to head of 'q'.This new vector is the resultant vector 'r'.
This sum of vector p (vector) and 'q' is vector.
i.e., 'r' (vector) = 'p'(vector) + 'q' (vector)
According to triangle law of vectors,if three vector are represented by the three sides of a triangle taken in order,than their resultant is zero. Thus if three vectors 'p'(vector),'q'(vector)and 't' can be represented completely in magnitude and direction by the three sides of triangle taken in order,then their vector sum is zero.
Therefore 'p'(vector)+'q'(vector)+'t'(vector) = 0.
The resultant _ 't'vector of 'p' (vector) and 'q'(vector) cancels the third vector 't'.
For parallelogram law and polygon law,pleas see the topic on parallelogram law of forces and polygon law of forces respectively in the forth coming topic composition of forces.
(iii) Subtraction of vectors
The process of subtracting one vector from another vector is equivalent to adding vectorially the negative of the vector to be subtracted. If 'p' (vector) and 'q' (vector) are two vectors,then
'p'(vector)-'q'(vector) = 'p'(vector) +(-q)(vector).
(iv) Multiplication of vectors
Multiplication of vectors can be of the following three kinds:
(a) Multiplication of vector by a scalar : The multiplication of vector by a scalar or pure number n gives a vector in the same direction but n times its magnitude.As an example multiplication of a vector 'p'(vector) with a number n gives a vector 'r'(vector),such that
→ →
'r' = n'p'
In case n is physical scalar quantity having units,then the units of 'r'(vector) will be changed and will not be the same as that of 'p'(vector).As for example if vector quantity velocity is multiplied by a scalar quantity mass.Then,the product is momentum,a different vector quantity having different units. In case the scalar is negative,the direction of the resultant vector is reversed.
(b) Scalar product or dot product of two vectors : The scalar product of two vectors is defined as the product of the magnitudes of the two vector and the cosine of the smaller angle between them.
It is also called as the dot product and is represented by a dot between the two vectors.
→ → → →
Therefore 'p' . 'q' = 'p' . 'q' cos 0 = 's'
S being a scalar quantity.
Again the dot product can be written as
→ → → →
'p' . 'q' = 'p' (Q cos 0)
→ → → →
'p' . 'q' = 'q' (P cos 0)
Thus scalar product of two vectors may be defined as the product of one vector with the projection (vector×cos 0) of the other vector along the direction of the first vector.
→ →
Special cases : (i) if 0 = 0° i.e., 'p' and 'q' are parallel to each other,then
'p'(vector) 'q'(vector) cos 0 = 'p' 'q' cos 0° = 'p' 'q'
Hence when two vectors are parallel to each other their dot product is equal to the product of their magnitudes.
(ii) If 0 = 90°, i.e., 'p' (vector) 'q'(vector) are perpendicular to each other,then
'p'(vector) 'q' = 'p''q' cos 0 = 'p''q' cos 90° = zero
Hence when two vector are perpendicular to each other then their dot product is zero.
It follows that i(perpendicular vector)×i(perpendicular vector) =j(perpendicular vector)×j = (1) (1) cos 0 = 1
and i(perpendicular vector)×(perpendicular vector)j =j(perpendicular vector)×i(perpendicular vector) = k(perpendicular vector)×j(perpendicular vector) = i(perpendicular vector)×k(perpendicular vector) = k(perpendicular vector)×i(perpendicular vector) = 0
Because angle between i(perpendicular vector)×i(perpendicular vector),j(perpendicular vector)
×j(perpendicular vector) and k(perpendicular vector)×k(perpendicular vector) is 0° and angle between i(perpendicular vector) and j(perpendicular vector) and j(perpendicular vector)and k(perpendicular vector) is 90°.
Properties of scalar product or dot product
1. Dot product of two vectors obey commutative law.
i.e., → → → →
P×Q = Q×P
2. Dot product of vector with itself is a scalar but the magnitude will be square of the magnitude of given vector.
i.e., → →
P×P = P×P cos 0°= P×P = P2
3. Dot product of two mutually perpendicular vector is zero.
i.e., → →
P×Q = P×Q cos 90° = 00 = 90°,cos 90° = 0
4. Dot product of two collinear vector is numerically equal to the product of their magnitudes.
i.e., → →
P×Q = P×Q cos 0o = P×Q
5. Dot product of vector obey distributive law.
i.e., → → → → → → →
P×(Q+R) = P×Q + P×R
6. Dot product of two vector is -ve when angle between them is greater than 90°.
i.e., → →
P×Q = -ve when o> 90°.
7. Dot product of unit vector
i(perpendicular vector)×i(perpendicular vector) = j(perpendicular vector)×j(perpendicular vector)= k(perpendicular vector)×k(perpendicular vector) = 1 as angle between them is 0o.
i(perpendicular vector)×j(perpendicular vector) =j(perpendicular vector)×k(perpendicular vector) =
k(perpendicular vector)×i(perpendicular vector) = 0 as angle between them is 90o.
8.If P(vector) = Px i(perpendicular vector)+Py j(perpendicular vector)+Pzk(perpendicular vector) and Q×(vector)=Qx i(perpendicular vector) + Py j(perpendicular vector)+ Pz k(perpendicular vector)
Then →→
P×Q = PxQx+PyQy + PzQz
Examples of some important relation of dot product
1.Work is dot product of force and displacement
'w' = 'f '(vector)*'s'(vector) = 'f '*'s' cos 0
2. Power is dot product of force and velocity
'p' = 'f '* 'v'= 'f '*'v' cos 0
Since vectors possess direction in addition to the magnitude as possessed by scalar quantities. So they can not be added by using law of algebra.Let us discuss the addition of two vectors by taking example of displacement vectors.
(a) When two vector act in the same direction :
Let a particle is displaced through a distance of 5m along west to east direction and let it be represented in magnitude and direction by p,let this particle be again displaced through 3m along same direction i.e.,west to east,let it be represented by 'q' then the resultant displacement of two vectors will be
→ →
P + Q = 8 m along west to east direction.
(b) When two vector are in opposite direction :
Let these two displacement vectors 'p'(vector) and 'q'(vector) be in opposite direction p (vector) is acting in west to two displacement vector will be 2m,but it will act in direction along which vector with greater magnitude acts i.e., along west to east direction.
(c) When two or more vectors act at some angle to each other : In this case,we make use of either (a) triangle law,(b) parallelogram law or (c) polygon law of vectors.Let us study them one by one.
(ii) Triangle law of vectors
If two vector can be represented both in magnitude and direction by the sides of a triangle taken in the same order,then the resultant is represented completely in magnitude and direction,by the third side of the triangle taken in the opposite order.
Suppose we have to add two vectors 'p' and 'q'.Place the initial point (tail) of 'q' on the terminal point (head) of p.Complete the triangle to get a new vector 'p'+'q' (vector) starting from tail of p to head of 'q'.This new vector is the resultant vector 'r'.
This sum of vector p (vector) and 'q' is vector.
i.e., 'r' (vector) = 'p'(vector) + 'q' (vector)
According to triangle law of vectors,if three vector are represented by the three sides of a triangle taken in order,than their resultant is zero. Thus if three vectors 'p'(vector),'q'(vector)and 't' can be represented completely in magnitude and direction by the three sides of triangle taken in order,then their vector sum is zero.
Therefore 'p'(vector)+'q'(vector)+'t'(vector) = 0.
The resultant _ 't'vector of 'p' (vector) and 'q'(vector) cancels the third vector 't'.
For parallelogram law and polygon law,pleas see the topic on parallelogram law of forces and polygon law of forces respectively in the forth coming topic composition of forces.
(iii) Subtraction of vectors
The process of subtracting one vector from another vector is equivalent to adding vectorially the negative of the vector to be subtracted. If 'p' (vector) and 'q' (vector) are two vectors,then
'p'(vector)-'q'(vector) = 'p'(vector) +(-q)(vector).
(iv) Multiplication of vectors
Multiplication of vectors can be of the following three kinds:
(a) Multiplication of vector by a scalar : The multiplication of vector by a scalar or pure number n gives a vector in the same direction but n times its magnitude.As an example multiplication of a vector 'p'(vector) with a number n gives a vector 'r'(vector),such that
→ →
'r' = n'p'
In case n is physical scalar quantity having units,then the units of 'r'(vector) will be changed and will not be the same as that of 'p'(vector).As for example if vector quantity velocity is multiplied by a scalar quantity mass.Then,the product is momentum,a different vector quantity having different units. In case the scalar is negative,the direction of the resultant vector is reversed.
(b) Scalar product or dot product of two vectors : The scalar product of two vectors is defined as the product of the magnitudes of the two vector and the cosine of the smaller angle between them.
It is also called as the dot product and is represented by a dot between the two vectors.
→ → → →
Therefore 'p' . 'q' = 'p' . 'q' cos 0 = 's'
S being a scalar quantity.
Again the dot product can be written as
→ → → →
'p' . 'q' = 'p' (Q cos 0)
→ → → →
'p' . 'q' = 'q' (P cos 0)
Thus scalar product of two vectors may be defined as the product of one vector with the projection (vector×cos 0) of the other vector along the direction of the first vector.
→ →
Special cases : (i) if 0 = 0° i.e., 'p' and 'q' are parallel to each other,then
'p'(vector) 'q'(vector) cos 0 = 'p' 'q' cos 0° = 'p' 'q'
Hence when two vectors are parallel to each other their dot product is equal to the product of their magnitudes.
(ii) If 0 = 90°, i.e., 'p' (vector) 'q'(vector) are perpendicular to each other,then
'p'(vector) 'q' = 'p''q' cos 0 = 'p''q' cos 90° = zero
Hence when two vector are perpendicular to each other then their dot product is zero.
It follows that i(perpendicular vector)×i(perpendicular vector) =j(perpendicular vector)×j = (1) (1) cos 0 = 1
and i(perpendicular vector)×(perpendicular vector)j =j(perpendicular vector)×i(perpendicular vector) = k(perpendicular vector)×j(perpendicular vector) = i(perpendicular vector)×k(perpendicular vector) = k(perpendicular vector)×i(perpendicular vector) = 0
Because angle between i(perpendicular vector)×i(perpendicular vector),j(perpendicular vector)
×j(perpendicular vector) and k(perpendicular vector)×k(perpendicular vector) is 0° and angle between i(perpendicular vector) and j(perpendicular vector) and j(perpendicular vector)and k(perpendicular vector) is 90°.
Properties of scalar product or dot product
1. Dot product of two vectors obey commutative law.
i.e., → → → →
P×Q = Q×P
2. Dot product of vector with itself is a scalar but the magnitude will be square of the magnitude of given vector.
i.e., → →
P×P = P×P cos 0°= P×P = P2
3. Dot product of two mutually perpendicular vector is zero.
i.e., → →
P×Q = P×Q cos 90° = 00 = 90°,cos 90° = 0
4. Dot product of two collinear vector is numerically equal to the product of their magnitudes.
i.e., → →
P×Q = P×Q cos 0o = P×Q
5. Dot product of vector obey distributive law.
i.e., → → → → → → →
P×(Q+R) = P×Q + P×R
6. Dot product of two vector is -ve when angle between them is greater than 90°.
i.e., → →
P×Q = -ve when o> 90°.
7. Dot product of unit vector
i(perpendicular vector)×i(perpendicular vector) = j(perpendicular vector)×j(perpendicular vector)= k(perpendicular vector)×k(perpendicular vector) = 1 as angle between them is 0o.
i(perpendicular vector)×j(perpendicular vector) =j(perpendicular vector)×k(perpendicular vector) =
k(perpendicular vector)×i(perpendicular vector) = 0 as angle between them is 90o.
8.If P(vector) = Px i(perpendicular vector)+Py j(perpendicular vector)+Pzk(perpendicular vector) and Q×(vector)=Qx i(perpendicular vector) + Py j(perpendicular vector)+ Pz k(perpendicular vector)
Then →→
P×Q = PxQx+PyQy + PzQz
Examples of some important relation of dot product
1.Work is dot product of force and displacement
'w' = 'f '(vector)*'s'(vector) = 'f '*'s' cos 0
2. Power is dot product of force and velocity
'p' = 'f '* 'v'= 'f '*'v' cos 0