To solve this problem, we need to find the volume of a parallelepiped that is formed by the vectors a = 3i + 4j + 9k, b = 2j + 5k, and c = 10k. To calculate it, we follow these steps:
Step 1: First, we calculate the cross product of vectors b and c. The cross product of two vectors b and c is given by the determinant of the 3x3 matrix that has i, j, and k on the first row, the components of b on the second row, and the components of c on the third row.
So, we find the cross product by evaluating the determinant:
i j k
0 2 5
0 0 10
The cross product of b and c (b x c) is given by i(20 - 0) - j(0 - 0) + k(0 - 0) = 20i.
Step 2: Now that we have the cross product of b and c, we find the dot product of vector a and the cross product (b x c). The dot product of two vectors a and (b x c) is equal to a1*(b x c)1 + a2*(b x c)2 + a3*(b x c)3 = 3*20 + 4*0 + 9*0 = 60.
Step 3: Lastly, we find the volume of the parallelepiped formed by vectors a, b, and c; it is given by the absolute value of the dot product found in step 2, which is still |60| = 60 cubic units.
So, the volume of the parallelepiped formed by the given vectors is 60 cubic units.