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1. Vectors and Geometry Let A = (0, 1, 3), B = (0, 6, 6), C = (3, −5,5), and O = (0, 0, 0). (a) Let u = AB, v = BỜ. (1) Simplify 2u +3v; (2) Find the angel /ABC; (3) Find the area of AABC.

1. Vectors and Geometry Let A = (0, 1, 3), B = (0, 6, 6), C = (3, −5,5), and O = (0, 0, 0). (a) Let u = AB, v = BỜ. (1) Simplify 2u +3v; (2) Find the angel /ABC; (3) Find the area of AABC.

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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# Educational Website Content ## 1. Vectors and Geometry **Given:** - Point \( A = (0, 1, 3) \) - Point \( B = (0, 6, 6) \) - Point \( C = (3, -5, 5) \) - Origin \( O = (0, 0, 0) \) ### Problems: (a) Let \( \mathbf{u} = \overrightarrow{AB}, \mathbf{v} = \overrightarrow{BC} \). 1. Simplify \( 2\mathbf{u} + 3\mathbf{v} \). 2. Find the angle \( \angle ABC \). 3. Find the area of triangle \( \triangle ABC \). (b) Find a vector equation for the line that passes through points \( A \) and \( B \); also find the distance from \( C \) to this line. (c) Find a linear equation for the plane that passes through points \( A \), \( B \), and \( C \); also determine a point \( D \) on the plane such that \( \overrightarrow{OD} \) is perpendicular to the plane. ## 2. Linear Equations ### Problems: (a) Let \( k \) be a constant. For unknowns \( x, y, z \), solve the system: - \( x + y + z = 1 \) - \( 2x + y + z = 5 \) - \( 6x + y + z = k \). (b) Let \( k \) be a constant. For unknowns \( x, y, z \), solve the system: - \( x + y + kz = 1 \) - \( x + ky + z = 1 \) - \( kx + y + z = -2 \). (c) Show that vectors \( \mathbf{u}, \mathbf{v}, \mathbf{w} \) are in the span of \(\{ \mathbf{u} + \mathbf{v}, 2\mathbf{u} + 3\mathbf{v}, 4\mathbf{v} + 6\mathbf{w} \}\). (d) Check the linear dependency of matrices \( B, C, D \) in the next problem. (e) Let - \( \mathbf{v}_1 = [
Transcribed Image Text:# Educational Website Content ## 1. Vectors and Geometry **Given:** - Point \( A = (0, 1, 3) \) - Point \( B = (0, 6, 6) \) - Point \( C = (3, -5, 5) \) - Origin \( O = (0, 0, 0) \) ### Problems: (a) Let \( \mathbf{u} = \overrightarrow{AB}, \mathbf{v} = \overrightarrow{BC} \). 1. Simplify \( 2\mathbf{u} + 3\mathbf{v} \). 2. Find the angle \( \angle ABC \). 3. Find the area of triangle \( \triangle ABC \). (b) Find a vector equation for the line that passes through points \( A \) and \( B \); also find the distance from \( C \) to this line. (c) Find a linear equation for the plane that passes through points \( A \), \( B \), and \( C \); also determine a point \( D \) on the plane such that \( \overrightarrow{OD} \) is perpendicular to the plane. ## 2. Linear Equations ### Problems: (a) Let \( k \) be a constant. For unknowns \( x, y, z \), solve the system: - \( x + y + z = 1 \) - \( 2x + y + z = 5 \) - \( 6x + y + z = k \). (b) Let \( k \) be a constant. For unknowns \( x, y, z \), solve the system: - \( x + y + kz = 1 \) - \( x + ky + z = 1 \) - \( kx + y + z = -2 \). (c) Show that vectors \( \mathbf{u}, \mathbf{v}, \mathbf{w} \) are in the span of \(\{ \mathbf{u} + \mathbf{v}, 2\mathbf{u} + 3\mathbf{v}, 4\mathbf{v} + 6\mathbf{w} \}\). (d) Check the linear dependency of matrices \( B, C, D \) in the next problem. (e) Let - \( \mathbf{v}_1 = [
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