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P Pankaj Sanodiya
vectors  will be  coplanar if  now, it is given that   now as we know that whenever two vectors are same in a vector triple product its value becomes zero,  so Hence when   is zero

P Pankaj Sanodiya
Given four points A = (3, 2, 1) , B= (4, x, 5), C= (4, 2, –2) and  D =(6, 5, –1) . These four points will be coplanar when the volume of the shape they make will be zero. which means  Hence the value of x is 5.

P Pankaj Sanodiya
Given four position vectors, And, now these will be coplanar when  Now, let's calculate the vector triple product of these vectors Hence, the four points are coplanar.

P Pankaj Sanodiya
Given these will be coplanar when, Hence the value of    is -2, irrespective of the value of . Hence  no value of   satisfies the condition of three vectors being coplanar.

P Pankaj Sanodiya
Given These will be coplanar when  Hence the value of for which vectors will be coplanar is 2.

P Pankaj Sanodiya
Given three vectors, These three will be coplanar when,

P Pankaj Sanodiya
Given  For coplanarity, So let's calculate the vector triple product of these three vectors, Hence the three vectors are coplanar.

P Pankaj Sanodiya
Given, Hence the value of  is 24.

P Pankaj Sanodiya
Given in the question  is the angle between any two vectors  and To find the value of Hence option D is correct.

P Pankaj Sanodiya
To find the value of   Hence option C is correct.

P Pankaj Sanodiya
Gicen in the question    be two unit vectors and   is the angle between them also Then is a unit vector if Hence option D is correct.

P Pankaj Sanodiya
Given   and  Now, let vector is inclined to  at  respectively. Now,  Now, Since,      Hence vector   is equally inclined to  .

P Pankaj Sanodiya
Given in the question  is the angle between two vectors  this will satisfy when  Hence option B is the correct answer.

P Pankaj Sanodiya
Given in the question,  are perpendicular and we need to prove that LHS=                                                                   if   are perpendicular,                                                                                                          = RHS  LHS ie equal to RHS Hence proved.

P Pankaj Sanodiya
Let, the sum of vectors  and  be  unit vector along  Now, the scalar product of this with  squaring both the side,

P Pankaj Sanodiya
Given, Let  now, since it is given that d is perpendicular to  and , we got the condition,    and       And        And  here we got 2 equation and 3 variable. one more equation will come from the condition:   so now we have three equation and three variable, On solving this three equation we get, , Hence Required vector : .

P Pankaj Sanodiya
Let a vector  is equally inclined to axis OX, OY  and OZ. let direction cosines of this vector be  Now  Hence direction cosines are: .

P Pankaj Sanodiya
Given, two adjacent sides of the parallelogram  The diagonal will be the resultant of these two vectors. so resultant R: Now unit vector in direction of R  Hence unit vector along the diagonal of the parallelogram  Now, Area of parallelogram  Hence the area of the parallelogram is .