Dot Product and Cross product
Dot product and cross product are two important operations in physics. Dot product is a scalar product and cross product is a vector product. In this article, we will discuss the difference between these two operations and their physical significance. Dot product is an operation that takes two vectors and returns a scalar quantity. The dot product of two vectors is equal to the product of their magnitudes and the cosine of the angle between them. Cross product is an operation that takes two vectors and returns a vector quantity. The cross product of two vectors shows direction of resultant vector in i j and k and the sine of the angle between them.
Dot Product:
When the
product of two vectors is scalar so the product is called dot product.
The dot
product is represented by a dot between two vectors.
The finest
example of dot product is work which is product of two vectors.
W = F.d
Where W is
scalar and force and displacement are vectors.
Using Dot product in Calculations:
Problem 1:
A = 2i + 3j - 2k
B = 4i + j +3k
Find the dot
product of vector A and B.
Note:
{
i . i = 1
j . j = 1
k . k = 1
Rest are
zero.
}
Solution:
A.B = (2i
+3j-2k) . (4i+j+3k)
A.B = 8(i.i)
+3(j.j) -6(k.k)
A.B = 8(1) +
3(1) - 6(1)
A.B = 8 + 3
– 6
A.B = 5
units
Problem 2:
A = 7i - j - 2k
B =
-5i - 3j +6k
Solution:
A.B = (7i -
j - 2k) . (-5i - 3j +6k)
A.B = -35(i.i)
+ 3(j.j) - 12(k.k)
A.B = -35(1)
+ 3(1) - 12(1)
A.B = -35 +
3 - 12
A.B = -44
units
When dot is removed the cosθ is added.
W = F.d
W = Fdcosθ
When the angle between two vector is zero or the two vector are parallel to each other so the dot product will be maximum.
W = Fdcosθ
W = Fdcos0°
W = Fd(1)
W = Fd
When the angle between two vector is 90° or the two vector are perpendicular to each other so the dot product will be minimum.
W = Fdcosθ
W = Fdcos90°
W = Fd(0)
W = 0
When the angle between two vectors is 180° or the vectors are anti parallel to each other so the dot product will be negative.
W = Fdcosθ
W = Fdcos180°
W = Fd(-1)
W = -Fd
Another Example:
Projection of vector A on B
Calculate the projection of vector A in direction of vector B.
A = 3i - 4j + 2k
B = 6i + 8j - 5k
Projection means shadow of vector A is on vector B.
Step 1: We will find unit vector of B which is a base.
B̂ = B / |B|
|B| = √a2 +b2
|B| = √ +b2
Cross Product:
When product of two vectors is a vector then the product is called cross product.
Cross product is represented by cross sign.
The best example of cross product is torque.
T = f x d
Where torque is vector as well as force and displacement are also vectors.
Using Cross Product in Calculations:
Cross product can be calculated by two ways:
By Matrix determinent
By direct method
Problem 1:
Find cross product of two vectors where vector
A = 3i - 5j + 4k
B = 7i - 10j - k
SOLUTION:
Step 1 : Write the coefficient of i j and k of both vectors in one matrix.
Step 2 : Get the determinent of the matrix.
i(ad-bc)-j(ad-bc)+k(ad-bc)
i(5+10)-j(-3-28)+k(-30+35)
i(15)-j(-31)+k(5)
15i+31j+5k
Ans;
Problem 2:
Find cross product of two vectors where vector
A = 6i + 3j + 8k
B = i- 2j + k
SOLUTION:
Step 1 : Write the coefficient of i j and k of both vectors in one matrix.
Step 2 : Get the determinent of the matrix.
i(ad-bc)-j(ad-bc)+k(ad-bc)
i(3+16)-j(6-8)+k(-12+3)
i(19)-j(-2)+k(9)
19i+2j+9k
Ans;
When cross is removed then the sinθ is added.
When two vectors are parallel there cross product will be minimum.
T = Fdsin0°
T = Fd(0)
T = 0
When the two vectors are perpendicular to each other then cross product will be maximum.
T = Fdsin90°
T = Fd(1)
T = Fd
