HOW TO FIND VECTOR PRODUCT OR CROSS PRODUCT OF TWO VECTORS

About "How to Find Vector Product or Cross Product of Two Vectors"

How to Find Vector Product or Cross Product of Two Vectors :

Here we are going to see how to find vector product or cross product of two vectors.

Working rule to find the cross product

Let 

Finding cross product of two vectors  - Examples

Question 1 :

Find the magnitude of a vector x b vector, if a vector  =  2i vector + j vector + 3k vector and b vector  =  3i vector+ 5j vector - 2k vector

Solution :

  =  i vector[-2 - 15] - j vector[-4 -9] + k vector[10-3]

  =  i vector[-17] - j vector[-13] + k vector[7]

a x b  =  -17 i vector + 13 j vector + 7 k vector

In order to find its magnitude, we have to take square root and find the sum of coefficients of i, j and k.

|a x b|  =  √(-17)2 + 132 + 72

  =  √(289 + 169 + 49)

  =  √507

Question 2 :

Show that 

Solution :

a x (b + c)  =  a x b + a x c  ---(1)

b x (c + a)  =  b x c + b x a  ---(2)

c x (a + b)  =  c x a + c x b  ---(3)

(1) + (2) + (3)

a x (b + c) + b x (c + a) + c x (a + b) 

=  a x b + a x c + b x c + b x a + c x a + c x b

Since commutative property is not applicable in cross product,

  =  a x b + a x c + b x c - a x b - a x c - b x c

  =  0

Hence it is proved.

Question 3 :

Find the vectors of magnitude 103 that are  perpendicular to the plane which contains i vector + 2j vector + k vector and i vector + 3j vector + 4k vector

Solution :

Let a vector  =  i vector + 2j vector + k vector

b vector  =  i vector + 3j vector + 4k vector

required vector perpendicular to given vectors 

=  ± μ [(a x b)/ |a x b|]

  =  i[8-3] - j[4-1] + k[3-2]

a x b  =  5i - 3j + k

|a x b|  =  √52 + (-3)2 + 12

  =  √(25+9+1)

  =  √35

Required vector  = ± (103/√35) (5i - 3j + k)

± (103/√35) (5i - 3j + k)

After having gone through the stuff given above, we hope that the students would have understood,"Properties of Scalar Product or Dot Product"

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