If in a right-angled triangle `A B C ,` the hypotenuse `A B=p` ,then `vec(AB).vec(AC)+ vec(BC). vec(BA)+ vec(CA).vec(CB)` is equal to `2p^2` b. `(p^2)/2` c. `p^2` d. none of these

If D is the midpoint of the side BC of a triangle ABC, prove that vec(AB)+vec(AC)=2vec(AD)

If D and E, are the midpoints of the sides AB and AC of a triangle ABC, prove that vec(BE) +vec(DC)=(3)/(2)vec(BC).

vec ba n d vec c are unit vectors. Then for any arbitrary vector vec a ,((( vec axx vec b)+( vec axx vec c))xx( vec bxx vec c)).( vec b- vec c) is always equal to a. | vec a| b. 1/2| vec a| c. 1/3| vec a| d. none of these

If vec A, vec B and vec C are vectors such that |vec B|=|vec C| . Prove that [(vec A+ vec B)xx (vec A + vec C)]xx (vec B+vec C).(vec B+ vec C)=0

If vec a is parallel to vec bxx vec c , then ( vec axx vec b)dot( vec axx vec c) is equal to a. | vec a|^2( vec b . vec c) b. | vec b|^2( vec a . vec c) c. | vec c|^2( vec a . vec b) d. none of these

The scalar vec Adot ( ( vec B+ vec C)xx( vec A+ vec B+ vec C)) equals a. 0 b. [ vec A vec B vec C]+[ vec B vec C vec A] c. [ vec A vec B vec C] d. none of these

If in triangle A B C , vec A B= vec u/(| vec u|)- vec v/(| vec v|)a n d vec A C=(2 vec u)/(| vec u|),w h e r e| vec u|!=| vec v|, then 1+cos2A+cos2B+cos2C=0 b. sinA=cos C c. projection of A C on B C is equal to B C d. projection of A B on B C is equal to A B

If P is any arbitrary point on the circumcirlce of the equllateral trangle of side length l units, then | vec P A|^2+| vec P B|^2+| vec P C|^2 is always equal to 2l^2 b. 2sqrt(3)l^2 c. l^2 d. 3l^2

If vec a , vec b , vec c are unit vectors such that vec adot vec b=0= vec adot vec c and the angle between vec ba n d vec c is pi//3 , then the value of | vec axx vec b- vec axx vec c| is 1//2 b. 1 c. 2 d. none of these

Distance of point P( vec p) from the plane vec rdot vec n=0 is a. | vec pdot vec n| b. (| vec pxx vec n|)/(| vec n|) c. (| vec pdot vec n|)/(| vec n|) d. none of these