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Exercise Set 1: Write the matrix A from the given vectors. Calculate the area (or volume) of the parallelogram (parallelepiped) defined by them. (a) ū= (1,4), v = (5,0) (b) u = (3,2), v = (6,—4) (c) u = (2,5, 0), v = (3,−2, −1), w = (−1, 4, 3) (d) u = the vector starting at the point (2, 4) and ending at the point (5, 11),

Exercise Set 1: Write the matrix A from the given vectors. Calculate the area (or volume) of the parallelogram (parallelepiped) defined by them. (a) ū= (1,4), v = (5,0) (b) u = (3,2), v = (6,—4) (c) u = (2,5, 0), v = (3,−2, −1), w = (−1, 4, 3) (d) u = the vector starting at the point (2, 4) and ending at the point (5, 11),

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Area Theorem:** If \( A \) is a 2 x 2 matrix, the area of the parallelogram determined by the columns of \( A \) is the absolute value of the determinant of \( A \), Area = \(|\text{det} (A)|\). The same formula calculates the volume of the parallelepiped determined by the columns of \( A \), if \( A \) is a 3 x 3 matrix. **Example:** The vectors \(\vec{u} = (2, 1)\), \(\vec{v} = (3, 0)\) define the parallelogram. *Diagram Explanation:* The graph shows a grid with vectors \(\vec{u}\) and \(\vec{v}\). Vector \(\vec{u}\) starts at the origin and points to (2,1), while vector \(\vec{v}\) points to (3,0). These vectors form a parallelogram shaded in blue on the graph. The matrix \( A = \begin{bmatrix} 2 & 3 \\ 1 & 0 \end{bmatrix} \), \(\text{det} A = -3\), \(|\text{det}(A)| = |-3| = 3\). --- **Exercise Set 1:** Write the matrix \( A \) from the given vectors. Calculate the area (or volume) of the parallelogram (parallelepiped) defined by them. (a) \(\vec{u} = (1, 4)\), \(\vec{v} = (5, 0)\) (b) \(\vec{u} = (3, 2)\), \(\vec{v} = (6, -4)\) (c) \(\vec{u} = (2, 5, 0)\), \(\vec{v} = (3, -2, -1)\), \(\vec{w} = (-1, 4, 3)\) (d) \(\vec{u}\) = the vector starting at the point (2, 4) and ending at the point (5, 11), \(\vec{v}\) = the vector starting at the point (2, 4) and ending at the point (-4, 5). For this part include a graph of the
Transcribed Image Text:**Area Theorem:** If \( A \) is a 2 x 2 matrix, the area of the parallelogram determined by the columns of \( A \) is the absolute value of the determinant of \( A \), Area = \(|\text{det} (A)|\). The same formula calculates the volume of the parallelepiped determined by the columns of \( A \), if \( A \) is a 3 x 3 matrix. **Example:** The vectors \(\vec{u} = (2, 1)\), \(\vec{v} = (3, 0)\) define the parallelogram. *Diagram Explanation:* The graph shows a grid with vectors \(\vec{u}\) and \(\vec{v}\). Vector \(\vec{u}\) starts at the origin and points to (2,1), while vector \(\vec{v}\) points to (3,0). These vectors form a parallelogram shaded in blue on the graph. The matrix \( A = \begin{bmatrix} 2 & 3 \\ 1 & 0 \end{bmatrix} \), \(\text{det} A = -3\), \(|\text{det}(A)| = |-3| = 3\). --- **Exercise Set 1:** Write the matrix \( A \) from the given vectors. Calculate the area (or volume) of the parallelogram (parallelepiped) defined by them. (a) \(\vec{u} = (1, 4)\), \(\vec{v} = (5, 0)\) (b) \(\vec{u} = (3, 2)\), \(\vec{v} = (6, -4)\) (c) \(\vec{u} = (2, 5, 0)\), \(\vec{v} = (3, -2, -1)\), \(\vec{w} = (-1, 4, 3)\) (d) \(\vec{u}\) = the vector starting at the point (2, 4) and ending at the point (5, 11), \(\vec{v}\) = the vector starting at the point (2, 4) and ending at the point (-4, 5). For this part include a graph of the
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