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Given in the question  is the angle between any two vectors  and To find the value of Hence option D is correct.
To find the value of   Hence option C is correct.
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.  
Given   and  Now, let vector is inclined to  at  respectively. Now,  Now, Since,      Hence vector   is equally inclined to  .
Given in the question  is the angle between two vectors  this will satisfy when  Hence option B is the correct answer.
Given in the question,  are perpendicular and we need to prove that LHS=                                                                   if   are perpendicular,                                                                                                          = RHS  LHS ie equal to RHS Hence proved.
Let, the sum of vectors  and  be  unit vector along  Now, the scalar product of this with  squaring both the side,
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 : .
Let a vector  is equally inclined to axis OX, OY  and OZ. let direction cosines of this vector be  Now  Hence direction cosines are: .
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 .
Given, two vectors  the point  R which divides line segment PQ in ratio 1:2 is given by  Hence position vector of R is . Now, Position vector of the midpoint of RQ which is the position vector of Point P . Hence, P is the mid-point of RQ
Given in the question, points A(1, – 2, – 8), B(5, 0, –2) and C(11, 3, 7) now let's calculate the magnitude of the vectors As we see that AB = BC + AC, we conclude that three points are colinear. we can also see from here, Point B divides AC in the ratio  2 : 3.    
Given in the question, Now, let vector Now, a unit vector in direction of  Now, A unit vector parallel to     OR
Given two vectors  Resultant of  and : Now, a unit vector in the direction of  Now, a unit vector of magnitude in direction of   Hence the required vector is .  
Given in the question, a unit vector,  We need to find the value of x The value of x   is  
No, if then we can not conclude that   . the condition     satisfies in the triangle. also, in a triangle,   Since, the condition  is contradicting with the triangle inequality, if  then we can not conclude that  
As the girl walks 4km towards west Position vector =    Now as she moves 3km in direction 30 degree east of north. hence final position vector is;
Given in the question  And we need to finrd the scalar components and magnitude of the vector joining the points P and Q Magnitiude of vector PQ Scalar components are
As we know  a unit vector in XY-Plane making an angle  with x-axis : Hence for  Answer- the unit vector in XY-plane, making an angle of with the positive direction of x-axis is
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