Question

Vectors A, B, and C are three non-coplanar unit vectors, each inclined with each other at an angle of 30 degrees. Then, the volume of the tetrahedron whose edges are A, B, and C is:

a) √3/4​​
b) √3/2​​
c) 3√3/4​​
d) √3​

Answer 1

The volume of the tetrahedron with edges represented by unit vectors A, B, and C, inclined at 30 degrees to each other, is given by the scalar triple product and is 3√3/4. Option C is correct.

Step-by-step explanation:

To find the volume of the tetrahedron with edges represented by three non-coplanar unit vectors A, B, and C, each inclined at an angle of 30 degrees to each other, we can use the scalar triple product. The scalar triple product (B x C).A gives the volume of the parallelepiped formed by vectors A, B, and C. Since all three vectors are unit vectors and inclined at an angle of 30 degrees, we can use the following formula for the volume V of the tetrahedron:

V = (1/6) * |(B x C).A|

Using the property that the magnitude of the cross product of two unit vectors at a 30-degree angle to each other is sin(30), and the dot product with another unit vector at a 30-degree angle to both is cos(30), we have:

V = (1/6) * |sin(30) * cos(30)|

Substituting the values for sin(30) and cos(30), we find:

V = (1/6) * |(1/2) * (√3/2)|

V = (√3/24) = c) 3√3/4