Find `lambda in R` such that `|quad lambda vec a|=1`, `vec a` being a non-zero vector.

If vec a and vec b are two vectors such that |quad vec a + vec b|= |quad vec a| , then prove that 2 vec a+ vec b is perpendicular to the vector vec b .

If vec a and vec b are two vectors such that |quad vec a| =10 , |quad vec b|=15 and vec a.vec b=75 sqrt 2 , find the angle between vec a and vec b .

In a four-dimensional space where unit vectors along the axes are hat i , hat j , hat k and hat l , and vec a_1, vec a_2, vec a_3, vec a_4 are four non-zero vectors such that no vector can be expressed as a linear combination of others and (lambda-1)( vec a_1- vec a_2)+mu( vec a_2+ vec a_3)+gamma( vec a_3+ vec a_4-2 vec a_2)+ vec a_3+delta vec a_4=0, then

If vec a and vec b are two vectors such that |quad vec a|=2 , |quad vec b|=1 and vec a. vec b=1 , then find the value of (3 vec a - 5 vec b).(2 vec a+7 vec b) .

If vectors vec a and vec b , are such that |vec a| =3,|vec b| =2/3 and vec a xx vec b is a unit vector, then find the angle between vec a and vec b .

If vec a=2 hat i+2 hat j+3 hat k , vec b=-hat i+2 hat j+hat k and vec c=3 hat i+hat j be such that vec a+lambda vec b is at right angles to vec c , then find lambda .

If veca is a non-zero vector of magnitude 'a' and 'lambda' is a non-zero scalar, such that |lambdaveca|=1 , then :

Find lambda if the vectors vec a=hat i-lambda hat j+3 hat k and vec b=4 hat i-5 hat j+2 hat k are perpendicular to each other.

Show that |vec a|vec b + |vec b|vec a perpendicular to |vec a|vec b - |vec b|vec a , for any two non-zero vectors vec a and vec b .

Let vec a , vec b , vec c be three vectors such that |quad vec a|=3 , |quad vec b|=4 and |quad vec c|=5 and one of them being perpendicular to the sum of the other two, find |quad vec a+vec b+vec c| .