Question

This exercise should be done two ways: by hand and using technology where possible. let a = 3 −3 0 9 0 −9 , b = 3 0 −1 9 −1 1 , c = x 1 w z r 4 . Evaluate the following expression: a⋅(b×c). Perform the calculation both manually and using technology where applicable. Show your steps and methodology for a comprehensive understanding of the process.

Answer 1

The result of the expression a⋅(b×c) is [54x - 3w + 9z + 81r -36x + 3w - 9z - 108r 0 - 9x + 9w + 0 - 81r].

Step-by-step explanation:

To evaluate a⋅(b×c), we first need to compute the cross product of vectors b and c. After calculating b×c, we then proceed to multiply the resulting vector by matrix a. Utilizing the cross product formula, the resultant vector [54x - 3w + 9z + 81r -36x + 3w - 9z - 108r 0 - 9x + 9w + 0 - 81r] is obtained by applying matrix multiplication between a and the vector obtained from the cross product of b and c. This is achieved by multiplying each element of matrix a by the corresponding elements of the vector, leading to the final computed expression.

The procedure begins with the calculation of the cross product of vectors b and c, followed by multiplying this resultant vector by matrix a to produce the final vector. The final result encapsulates the outcome of these computations, showcasing the linear combination of elements from matrices a, b, and c according to the principles of matrix and vector multiplication. This process is foundational in linear algebra, enabling the determination of resultant vectors through systematic multiplication and addition of their corresponding elements.