The three vectors hat i + hat j , hat j + hat k , hat k + hat i taken two at a time form three planes. The three unit vectors drawn perpendicular to these three planes form a parallelopiped of volume :

Question

The three vectors hat i + hat j , hat j + hat k , hat k + hat i taken two at a time form three planes. The three unit vectors drawn perpendicular to these three planes form a parallelopiped of volume : (A) (1/3) (B) 4 (C) (3 √3/4) (D) (4/3 √3)

Tardigrade Admin · Accepted Answer

( hati+ hatj) ×( hatj+ hatk)= hati- hatj+ hatk ; so the unit vector perp to the plane of hat i + hat j and hat j + hat k is (1/√3)( hat i - hat j + hat k ) ⋅ Similarly, the other two unit vectors are (1/√3)( hat i + hat j - hat k ) and (1/√3)(- hat i + hat j + hat k ) Hence, the required volume =(1/3 √3)| beginarrayccc1 -1 1 1 1 -1 -1 1 1 endarray|=(4/3 √3)

Q. The three vectors $\hat{i} + \hat{j}, \hat{j} + \hat{k}, \hat{k} + \hat{i}$ taken two at a time form three planes. The three unit vectors drawn perpendicular to these three planes form a parallelopiped of volume :

$\frac{1}{3}$

$4$

$\frac{3 3}{4}$

$\frac{4}{3 3}$

Q. The three vectors i^+j^​,j^​+k^,k^+i^ taken two at a time form three planes. The three unit vectors drawn perpendicular to these three planes form a parallelopiped of volume :

31​

4

433​​

33​4​

Q. The three vectors $\hat{i} + \hat{j}, \hat{j} + \hat{k}, \hat{k} + \hat{i}$ taken two at a time form three planes. The three unit vectors drawn perpendicular to these three planes form a parallelopiped of volume :

$\frac{1}{3}$

$4$

$\frac{3 3}{4}$

$\frac{4}{3 3}$