2. Linear Equations (a) Let k be a constant. For unknown 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 unknown x, y, z, solve the system x + y + kz ky + z = : 1, kx + y + z = -2. (c) Show that u, v, w is in span{u + v, 2u + 3v, 4v +6w}. = 1,x+
2. Linear Equations (a) Let k be a constant. For unknown 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 unknown x, y, z, solve the system x + y + kz ky + z = : 1, kx + y + z = -2. (c) Show that u, v, w is in span{u + v, 2u + 3v, 4v +6w}. = 1,x+
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
Problem number 2 please. Thanks for the help.
![## Vectors and Geometry
Let \( A = (0, 1, 3), B = (0, 6, 6), C = (3, -5, 5) \), and \( O = (0, 0, 0) \).
1. **(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 \( \triangle ABC \).
2. **(b)**
- Find a vector equation for the line that passes through \( A \) and \( B \); also find the distance from \( C \) to the line.
3. **(c)**
- Find a linear equation for the plane that passes through \( A, B, \) and \( C \); also find a point \( D \) on the plane such that \( \overrightarrow{OD} \) is perpendicular to the plane.
## Linear Equations
1. **(a)**
- Let \( k \) be a constant. For unknown \( x, y, z \), solve the system:
\[
\begin{align*}
x + y + z &= 1, \\
2x + y + z &= 5, \\
6x + y + z &= k.
\end{align*}
\]
2. **(b)**
- Let \( k \) be a constant. For unknown \( x, y, z \), solve the system:
\[
\begin{align*}
x + y + kz &= 1, \\
x + ky + z &= 1, \\
kx + y + z &= -2.
\end{align*}
\]
3. **(c)**
- Show that \( \mathbf{u}, \mathbf{v}, \mathbf{w} \) is in span\(\{ \mathbf{u} + \mathbf{v}, 2\mathbf{u} + 3\mathbf{v}, 4\mathbf{v} +](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcf673b33-84d3-4207-a3d8-77b439e8ab65%2Fe4a9dbd9-b9ad-4c16-997a-a4be7622c7d4%2Fbowufds_processed.png&w=3840&q=75)
Transcribed Image Text:## Vectors and Geometry
Let \( A = (0, 1, 3), B = (0, 6, 6), C = (3, -5, 5) \), and \( O = (0, 0, 0) \).
1. **(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 \( \triangle ABC \).
2. **(b)**
- Find a vector equation for the line that passes through \( A \) and \( B \); also find the distance from \( C \) to the line.
3. **(c)**
- Find a linear equation for the plane that passes through \( A, B, \) and \( C \); also find a point \( D \) on the plane such that \( \overrightarrow{OD} \) is perpendicular to the plane.
## Linear Equations
1. **(a)**
- Let \( k \) be a constant. For unknown \( x, y, z \), solve the system:
\[
\begin{align*}
x + y + z &= 1, \\
2x + y + z &= 5, \\
6x + y + z &= k.
\end{align*}
\]
2. **(b)**
- Let \( k \) be a constant. For unknown \( x, y, z \), solve the system:
\[
\begin{align*}
x + y + kz &= 1, \\
x + ky + z &= 1, \\
kx + y + z &= -2.
\end{align*}
\]
3. **(c)**
- Show that \( \mathbf{u}, \mathbf{v}, \mathbf{w} \) is in span\(\{ \mathbf{u} + \mathbf{v}, 2\mathbf{u} + 3\mathbf{v}, 4\mathbf{v} +
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