If `vec(x), vec(y), vec(z)` are three vectors, show that the points having positive vectors `7vec(x) - vec(z), vec(x) + 2vec(y) + 3vec(z) and -2vec(x) + 3vec(y) + 5vec(z)` are collinear

If vec(a), vec(b), vec(c) are three given vectors, show that the points having vectors 7vec(a)-vec(c), vec(a)+2vec(b) " and " -2vec(a)+3vec(b)+0.5vec(c) are collinear.

If vec(a) , vec(b)and vec(a) xx vec(b) are three unit vectors , find the angles beween the vectors vec(a) and vec(b)

For non-zero vectors vec(a) and vec(b), " if " |vec(a) + vec(b)| lt |vec(a) - vec(b)| , then vec(a) and vec(b) are-

If vec(p) is a unit vector and ( vec (x) - vec(p)). (vec(x)+vec(p)) = 8 , then find |vec (x)|

If vec a , vec b are two non-collinear vectors, prove that the points with position vectors vec a+ vec b , vec a- vec b and vec a+lambda vec b are collinear for all real values of lambdadot

For the vector vec(a) and vec (b) if |vec(a) + vec(b)| = |vec(a) - vec(b)| , show that vec(a) and vec (b) are perpendicular

Suppose that vec(p), vec(q) and vec(r) are three non-coplanar vectors in R. Let the components of a vector vec(s) along vec(p), vec(q) and vec(r) be 4, 3 and 5 respectively. If the components of this vector vec(s) " along" (-vec(p) + vec(q) + vec(r)), (vec(p) - vec(q) + vec(r)) and (-vec(p) - vec(q) + vec(r)) are x,y and z respectively, then the value of 2x + y + z is

Three non-zero vectors vec(a), vec(b), vec(c) are such that no two of them are collinear. If the vector (vec(a)+vec(b)) is collinear with the vector vec(c) and the vector (vec(b)+vec(c)) is collinear with vector vec(a) , then prove that (vec(a)+vec(b)+vec(c)) is a null vector.

Show that the vector vec(a) is perpendicular to the vector vec(b) - (vec(a).vec(b))/(|vec(a)|^(2))vec(a)

Let vec(a), vec(b) and vec(c) be three non-zero vectors such that no two of them are collinear and (vec(a) xx vec(b)) xx vec(c) = (1)/(3) |vec(b)||vec(c)| vec(a) . If theta is the angle between vector vec(b) and vec(c) , then a value of sin theta is