Examine the following vector for linear independence:
(1) `vec i+vec j+vec k ,2vec i+3vec j-veck,-vec i-2vecj+2vec k`
(2) `3veci+vec j-vec k, 2vec i-vec j+7 vec k, 7 veci-vec j+13vec k`

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 .

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)

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.

Given vec a=x hat i+y hat j+2 hat k , vec b= hat i- hat j+ hat k , vec c= hat i+2 hat j ; vec a_|_ vec b , vec adot vec c=4. Then [ vec a vec b vec c]^2=| vec a| b. [ vec a vec b vec c]^=| vec a| c. [ vec a vec b vec c]^=0 d. [ vec a vec b vec c]^=| vec a|^2

Statement 1: vec a=3 vec i+p vec j+3 vec k and vec b=2 vec i+3 vec j+q vec k are parallel vectors if p=9//2a n dq=2. Statement 2: if vec a=a_1 vec i+a_2 vec j+a_3 vec ka n d vec b=b_1 vec i+b_2 vec j+b_3 vec k are parallel, then (a_1)/(b_1)=(a_2)/(b_2)=(a_3)/(b_3)dot Which of the following Statements is/are correct ?

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

If aa n db are nonzero non-collinear vectors, then [ vec a vec b hat i] hat i+[ vec a vec b hat j] hat j+[ vec a vec b hat k] hat k is equal to a. vec axx vec b b. vec a+ vec b c. vec a- vec b d. vec bxx vec a

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+ hat j+ hat k and vec b= hat i-2 hat j+ hat k , then find vector vec c such that vec a . vec c=2 and vec axx vec c= vec bdot

Find a unit vector perpendicular to each of the vectors ( vec a+ vec b) and ( vec a- vec b) , where vec a= hat i+ hat j+ hat k , vec b= hat i+2 hat j+3 hat k .