If the projections of vector ` vec a` on `x` -, `y` - and `z` -axes are 2, 1 and 2 units ,respectively, find the angle at which vector ` vec a` is inclined to the `z` -axis.

Find the angle of vector vec a = 6 hat i + 2 hat j - 3 hat k with x -axis.

Suppose that vec p,vecqand vecr are three non- coplaner in R^(3) ,Let the components of a vector vecs along vecp , vec q and vecr be 4,3, and 5, respectively , if the components this vector vec s along (-vecp+vec q +vecr),(vecp-vecq+vecr) and (-vecp-vecq+vecr) are x, y and z , respectively , then the value of 2x+y+z is

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

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)

If hati and hatj are unit vectors along x and y axes respectively then the angle made by ( hati+hatj) vector with the x-axis is …….. [Fill in the blanks]

If the position coordinates of the points A and B are (x_1,y_1,z_1) and (x_2,y_2,z_2) respectively , determine the magnitude and direction of the vector vec(AB) .

If three unit vectors vec a , vec b , and vec c satisfy vec a+ vec b+ vec c=0, then find the angle between vec a and vec bdot

vec u , vec v and vec w are three non-coplanar unit vecrtors and alpha,beta and gamma are the angles between vec u and vec v , vec v and vec w ,a n d vec w and vec u , respectively, and vec x , vec y and vec z are unit vectors along the bisectors of the angles alpha,betaa n dgamma , respectively. Prove that [ vecx xx vec y vec yxx vec z vec zxx vec x]=1/(16)[ vec u vec v vec w]^2sec^2(alpha/2)sec^2(beta/2)sec^2(gamma/2) .

If two vectors vec(a) and vec (b) are such that |vec(a) . vec(b) | = |vec(a) xx vec(b)|, then find the angle the vectors vec(a) and vec (b)

Find the unit vector in the direction of vector vec (PQ) , where P and Q are the points (1,2,3) and (4,5,6) , respectively.