ABCDE is a pentagon .prove that the resultant of force ` vec A B ,` ` vec A E` ,` vec B C` ,` vec D C` ,` vec E D` and ` vec A C` ,is 3` vec A C` .

A B C D E is pentagon, prove that vec A B + vec B C + vec C D + vec D E+ vec E A = vec0 vec A B+ vec A E+ vec B C+ vec D C+ vec E D+ vec A C=3 vec A C

A , B , C , D are any four points, prove that vec A B dot vec C D+ vec B Cdot vec A D+ vec C Adot vec B D=0.

Prove that [ vec a, vec b, vec c + vec d] = [ vec a, vec b, vec c] + [ vec a, vec b , vec d] .

If vec a , vec ba n d vec c are non-coplanar vectors, prove that the four points 2 vec a+3 vec b- vec c , vec a-2 vec b+3 vec c ,3 vec a+ 4 vec b-2 vec ca n d vec a-6 vec b+6 vec c are coplanar.

Prove that if the vectors vec a, vec b, vec c satisfy vec a+ vec b + vec c = vec 0 , then vec bxx vec c = vec c xx vec a = vec a xx vec b

Prove that vec a xx (vec b + vec c) + vec b xx(vec a + vec c)+ vec c xx(vec a + vec b) = vec 0

For any three vectors veca, vec b, vec c prove that (vec a + vec b)+ vec c = vec a + (vec b + vec c)

If vec r . vec a= vec r . vec b= vec rdot vec c=1/2 or some nonzero vector vec r , then the area of the triangle whose vertices are A( vec a),B( vec b)a n dC( vec c)i s( vec a , vec b , vec c are non-coplanar ) a. |[ vec a vec b vec c]| b. | vec r| c. |[ vec a vec b vec c] vec r| d. none of these

Show that the vectors vec a, vec b and vec c are coplanar if vec a + vec b, vec b + vec c, vec c+ vec a are coplanar.

Vectors vec Aa n d vec B satisfying the vector equation vec A+ vec B= vec a , vec Axx vec B= vec ba n d vec A*vec a=1,w h e r e vec aa n d vec b are given vectors, are a. vec A=(( vec axx vec b)- vec a)/(a^2) b. vec B=(( vec bxx vec a)+ vec a(a^2-1))/(a^2) c. vec A=(( vec axx vec b)+ vec a)/(a^2) d. vec B=(( vec bxx vec a)- vec a(a^2-1))/(a^2)