[\begin{array}{c}-5 \\ 6 \\ 0\end{array}] \cdot[\begin{array}{ll}3 & -1\end{array}]
Answer (2)
Recognize that the given operation is a matrix multiplication.
Multiply the matrices: [ − 5 6 0 ] ⋅ [ 3 − 1 ] = [ − 5 ( 3 ) − 5 ( − 1 ) 6 ( 3 ) 6 ( − 1 ) 0 ( 3 ) 0 ( − 1 ) ]
Simplify the resulting matrix.
The final answer is: [ − 15 5 18 − 6 0 0 ]
Explanation
Understanding the Problem We are asked to compute the product of a column vector and a row vector. The column vector is a 3 × 1 matrix, and the row vector is a 1 × 2 matrix. The result of the multiplication will be a 3 × 2 matrix.
Performing the Matrix Multiplication To perform the matrix multiplication, we multiply each element of the column vector by each element of the row vector. This gives us:
[ − 5 6 0 ] ⋅ [ 3 − 1 ] = [ − 5 ( 3 ) − 5 ( − 1 ) 6 ( 3 ) 6 ( − 1 ) 0 ( 3 ) 0 ( − 1 ) ]
Simplifying the Result Now, we simplify the resulting matrix:
[ − 15 5 18 − 6 0 0 ]
Examples
Matrix multiplication is used extensively in computer graphics to perform transformations on 3D models. For example, rotating, scaling, and translating a 3D object can be achieved by multiplying the object's vertices (represented as vectors) by transformation matrices. This allows for efficient manipulation and rendering of complex scenes in video games, simulations, and other graphical applications.
To multiply the column vector − 5 6 0 with the row vector [ 3 − 1 ] , we create a 3x2 matrix where each element is the product of the corresponding elements of the two vectors. The resulting matrix is (\begin{bmatrix}-15 & 5 \\ 18 & -6 \\ 0 & 0\end{bmatrix}.
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