Find the volume of the box with the vectors A = 2i + j + k , B = 3i - 4j+ 2k , C = i - j+ 2k as edges.
That is just A•B×C
You would want
| A dot (B x C) |
= | <2,1,1> dot <-6,-4,1> | ......... I assume you know how to find cross and dot products
= | -12 - 4 + 1 | = 15 units^3
To find the volume of a box using vectors, you need to find the scalar triple product of the three vectors, which gives the signed volume of the parallelepiped formed by those vectors. The scalar triple product of three vectors, A, B, and C, is defined as:
V = A · (B × C)
Note that the dot operator (·) represents the dot product and the cross operator (×) represents the cross product.
Let's calculate the volume step by step:
1. Calculate the cross product of vectors B and C:
B × C = (3i - 4j + 2k) × (i - j + 2k)
To calculate the cross product, you can use the determinant of a 3x3 matrix:
B × C = [(b2c3 - b3c2)i - (b1c3 - b3c1)j + (b1c2 - b2c1)k]
Substituting the values of vectors B and C:
B × C = [(4*2 - 2*(-1))i - (3*2 - 2*1)j + (3*(-1) - 4*1)k]
= [10i - 4j - 7k]
2. Calculate the dot product of vector A with B × C:
A · (B × C) = (2i + j + k) · (10i - 4j - 7k)
To calculate the dot product, multiply the corresponding components of the vectors and sum them up:
A · (B × C) = 2*10 + 1*(-4) + 1*(-7)
= 20 - 4 - 7
= 9
3. Calculate the absolute value of the scalar triple product:
|V| = |A · (B × C)| = |9| = 9
Therefore, the volume of the box formed by the vectors A, B, and C is 9 cubic units.