- The input vector \(\begin{bmatrix} -7 \ 11 \ 2 \end{bmatrix}\) has 3 rows and 1 column, giving it a dimension of \(3 \times 1\).
- Scalar multiplication by \(3\) affects the values within the vector but does not change its dimensions.
- Multiplying the vector by \(3\) results in \(\begin{bmatrix} -21 \ 33 \ 6 \end{bmatrix}\), which still maintains the original \(3 \times 1\) structure.
- The correct size of the resulting product is \(\boxed{3 \times 1}\).
Explanation
- Analyzing the Input Vector
The given expression involves the scalar multiplication of a column vector by a scalar. Specifically, we are looking at the product:
\[ 3\left[\begin{array}{c} -7 \ 11 \ 2 \end{array}\right] \]
The vector inside the brackets, \(\begin{bmatrix} -7 \ 11 \ 2 \end{bmatrix}\), is a column vector with three elements arranged vertically, which means it has 3 rows and 1 column. Its dimensions are \(3 \times 1\).
- Understanding Scalar Multiplication and Dimensionality
Scalar multiplication is a fundamental operation in linear algebra where each element of a matrix (or vector) is multiplied by a given real number. A crucial property of scalar multiplication is that the dimensions of the matrix remain entirely unchanged after the operation. If a matrix \(A\) has dimensions \(m \times n\), then the product \(kA\), where \(k\) is a scalar, will also have dimensions \(m \times n\).
- Calculating the Product and Verifying Dimensions
By applying the scalar multiplication rule, we multiply each individual component of the vector by \(3\):
\[ 3\left[\begin{array}{c} -7 \ 11 \ 2 \end{array}\right] = \left[\begin{array}{c} 3 \times (-7) \ 3 \times 11 \ 3 \times 2 \end{array}\right] = \left[\begin{array}{c} -21 \ 33 \ 6 \end{array}\right] \]
The resulting vector \(\begin{bmatrix} -21 \ 33 \ 6 \end{bmatrix}\) still consists of three elements in a single column. Thus, it retains the dimensions \(3 \times 1\).
- Conclusion
Since the initial vector was of size \(3 \times 1\) and scalar multiplication preserves these dimensions, the product is also a \(3 \times 1\) vector. Comparing this result with the given options, we identify the correct size. Therefore, the product has a size of \(3 \times 1\).
Examples
Scalar multiplication is used everywhere in computer graphics and data science. For instance, if you have a 3D coordinate vector representing the position of an object in a virtual environment, multiplying that vector by a scalar can scale the object's distance from the origin. If you have the vector \(\mathbf{v} = \begin{bmatrix} 2 \ 5 \ 1 \end{bmatrix}\), scaling it by \(2\) gives \(\begin{bmatrix} 4 \ 10 \ 2 \end{bmatrix}\), which effectively doubles the position while maintaining the same dimensional structure of a \(3 \times 1\) column vector.