Example Question - dot product
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Matrix Multiplication Example
The image shows two matrices that we are supposed to multiply. Let's call the first matrix A and the second matrix B: Matrix A: \[ \begin{bmatrix} 3 & 2 & 5 \\ 2 & 3 & 1 \end{bmatrix} \] Matrix B: \[ \begin{bmatrix} 4 & 5 & -5 \\ 5 & -1 & 6 \\ 1 & 6 & -1 \end{bmatrix} \] To multiply these matrices, each element of the resulting matrix is calculated by taking the dot product of the corresponding row from the first matrix and the corresponding column from the second matrix. The resulting matrix will have the same number of rows as the first matrix (2 rows) and the same number of columns as the second matrix (3 columns). Let's calculate the elements of the resulting matrix one by one: Top-left element (C11) = (3 * 4) + (2 * 5) + (5 * 1) = 12 + 10 + 5 = 27 Top-middle element (C12) = (3 * 5) + (2 * -1) + (5 * 6) = 15 - 2 + 30 = 43 Top-right element (C13) = (3 * -5) + (2 * 6) + (5 * -1) = -15 + 12 - 5 = -8 Bottom-left element (C21) = (2 * 4) + (3 * 5) + (1 * 1) = 8 + 15 + 1 = 24 Bottom-middle element (C22) = (2 * 5) + (3 * -1) + (1 * 6) = 10 - 3 + 6 = 13 Bottom-right element (C23) = (2 * -5) + (3 * 6) + (1 * -1) = -10 + 18 - 1 = 7 Now we can construct the resulting matrix C: \[ \begin{bmatrix} 27 & 43 & -8 \\ 24 & 13 & 7 \end{bmatrix} \] So, the product of the two matrices A and B is: \[ \begin{bmatrix} 27 & 43 & -8 \\ 24 & 13 & 7 \end{bmatrix} \]
Determining Perpendicular Vectors by Dot Product
To determine whether two vectors are perpendicular, their dot product must equal zero. Given the vectors \( \vec{A} = k\hat{i} + 4\hat{j} \) and \( \vec{B} = -\hat{i} + 2\hat{j} \), the dot product of these two vectors is computed as: \( \vec{A} \cdot \vec{B} = (k\hat{i} + 4\hat{j}) \cdot (-\hat{i} + 2\hat{j}) \) This results in: \( \vec{A} \cdot \vec{B} = k(-1) + 4(2) \) \( \vec{A} \cdot \vec{B} = -k + 8 \) Since the vectors are perpendicular, their dot product is zero. Therefore: \( -k + 8 = 0 \) Solving for \( k \) gives us: \( k = 8 \) So the value of \( k \) is 8, which corresponds to option D.
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