Let veca = hati - hatk , vecb = x hati + hatj + (1 - x) hatk and vecc = y hati + x hatj + ( 1 +x -y ) hatk . Then [ veca , vecb , vecc ] depends on

Question

Let veca = hati - hatk , vecb = x hati + hatj + (1 - x) hatk and vecc = y hati + x hatj + ( 1 +x -y ) hatk . Then [ veca , vecb , vecc ] depends on (A) only y (B) only x (C) both x and y (D) neither x nor y

Tardigrade Admin · Accepted Answer

veca = hati - hatk , vecb = x hati + hatj + (1-x) hatk and vecc = y hati + x hatj + (1+ x -y) hatk [ veca vecb vecc] = veca vecb × vecc = |1&0&-1 x&1&1-x y&x&1+x-y| = 1[1+x-y-x+x2] - [x2 -y] = 1 - y + x2 -x2 + y = 1 Hence [ veca , vecb , vecc ] is independent of x and y both.

Q. Let a=i^−k^,b=xi^+j^​+(1−x)k^ and c=yi^+xj^​+(1+x−y)k^. Then [a,b,c] depends on

only y

only x

both x and y

neither x nor y

a=i^−k^,b=xi^+j^​+(1−x)k^ and c=yi^+xj^​+(1+x−y)k^ [abc]=ab×c=∣∣​1xy​01x​−11−x1+x−y​∣∣​ =1[1+x−y−x+x2]−[x2−y] =1−y+x2−x2+y =1 Hence [a,b,c] is independent of x and y both.

Q. Let $a = \hat{i} - \hat{k}, b = x \hat{i} + \hat{j} + (1 - x) \hat{k}$ and $c = y \hat{i} + x \hat{j} + (1 + x - y) \hat{k} .$ Then $[a, b, c]$ depends on