If the volume of a parallelopiped, whose coterminus edges are given by the vectors vec a = hat i + hat j + n hat k , vec b =2 hat i +4 hat j - n hat k and vec c = hat i + n hat j +3 hat k ( n ≥ 0), is 158 cu units, then :

Question

If the volume of a parallelopiped, whose coterminus edges are given by the vectors vec a = hat i + hat j + n hat k , vec b =2 hat i +4 hat j - n hat k and vec c = hat i + n hat j +3 hat k ( n ≥ 0), is 158 cu units, then : (A)  vec a ⋅ vec c =17 (B)  vec b ⋅ vec c =10 (C) n =7 (D) n =9

Tardigrade Admin · Accepted Answer

v =[ vec a vec b vec c ] 158 =|1&1&n 2&4&-n 1&n&3|, n ge 0 158=1(12+ n 2)-(6+ n )+ n (2 n -4) 158= n 2+12-6- n +2 n 2-4 n 3 n 2-5 n -152=0 n =8,-(38/6) (rejected) vec a ⋅ vec c =1+ n +3 n =1+4 n =33 vec b ⋅ vec c =2+4 n -3 n =2+ n =10

Q. If the volume of a parallelopiped, whose coterminus edges are given by the vectors $a = \hat{i} + \hat{j} + n \hat{k}, b = 2 \hat{i} + 4 \hat{j} - n \hat{k}$ and $c = \hat{i} + n \hat{j} + 3 \hat{k} (n \geq 0)$ , is $158 c u$ units, then :

$a \cdot c = 17$

$b \cdot c = 10$

$n = 7$

$n = 9$

$v = [a b c]$
$158 = ∣ ∣ 121 14 n n - n 3 ∣ ∣, n \geq 0$
$158 = 1 (12 + n^{2}) - (6 + n) + n (2 n - 4)$
$158 = n^{2} + 12 - 6 - n + 2 n^{2} - 4 n$
$3 n^{2} - 5 n - 152 = 0$
$n = 8, - \frac{38}{6}$ (rejected)
$a \cdot c = 1 + n + 3 n = 1 + 4 n = 33$
$b \cdot c = 2 + 4 n - 3 n = 2 + n = 10$

Q. If the volume of a parallelopiped, whose coterminus edges are given by the vectors a=i^+j^​+nk^,b=2i^+4j^​−nk^ and c=i^+nj^​+3k^(n≥0), is 158cu units, then :

a⋅c=17

b⋅c=10

n=7

n=9