Statement 1: let `A( vec i+ vec j+ vec k)a n dB( vec i- vec j+ vec k)` be two points. Then point `P(2 vec i+3 vec j+ vec k)` lies exterior to the sphere with `A B` as its diameter. Statement 2: If `Aa n dB` are any two points and `P` is a point in space such that ` vec P Adot vec P B >0` , then point `P` lies exterior to the sphere with `A B` as its diameter.

If vec a=2 vec i+3 vec j- vec k , vec b=- vec i+2 vec j-4 vec k a n d vec c= vec i+ vec j+ vec k , then find thevalue of ( vec axx vec b)dot(( vec axx vec c)dot)

Find a unit vector vec c if vec -i+vec j-vec k bisects the angle between vec c and 3 vec i+4vec j .

Statement 1:Let A( vec a),B( vec b)a n dC( vec c) be three points such that vec a=2 hat i+ hat k , vec b=3 hat i- hat j+3 hat ka n d vec c=- hat i+7 hat j-5 hat kdot Then O A B C is a tetrahedron. Statement 2: Let A( vec a),B( vec b)a n dC( vec c) be three points such that vectors vec a , vec ba n d vec c are non-coplanar. Then O A B C is a tetrahedron where O is the origin.

Let vec a= hat i+2 hat j+ hat k , vec b= hat i- hat j+ hat ka n d vec c= hat i+ hat j- hat kdot Then find [vec a vec b vec c]

Let vec a = vec i + vec j + vec k and let vec r be a variable vector such that vec r cdot vec i , vec r cdot vec j and vec r cdot vec k are positive integers. If vec r cdot vec a le 12 then the number of values of vec r is

If vec a.hat i= vec a.( hat i+ hat j)= vec a.( hat i+ hat j+ hat k) , then find the unit vector vec a

Let vec A=2 vec i+ vec k , vec B= vec i+ vec j+ vec kdot vec C = 4hati-3hatj+7hatk Determine a vector vec R satisfying vec Rxx vec B= vec Cxx vec B and vec R. vec A=0.

If alpha=2vec(i) +3vec(j) -5vec(k) and beta= vec(i)-vec(j) , then find the value of alpha . beta

The ratio in which the plane vec rdot (vec i-2 vec j+3 vec k ) =17 divides the line joining the points -2 vec i+4 vec j+7 vec k and 3 vec i-5 vec j+8 vec k is a. 1:5 b. 1: 10 c. 3:5 d. 3: 10

Prove that ( vec a.hat i)( vec axx hat i)+( vec a.j)( vec axx hat j)+( vec a. hat k)( vec axx hat k)=0.