In 3-D, let L be a line that does not pass through the origin. let v1, v2, and v3 be three vector that touch the line. Are the vectors {v1, v2, v3} linearly independent? Or dependent? ### Understanding 3D Coordinate Systems and Vector DecompositionThe diagram provided illustrates a 3D coordinate system consisting of three mutually perpendicular axes, labeled X, Y, and Z. In this coordinate system, we have several vectors, each with specific directions and magnitudes. **Axes:**1. **X-Axis**: Represents the horizontal axis in the 3D space.2. **Y-Axis**: Represents the horizontal axis perpendicular to the X-axis in the 3D space.3. **Z-Axis**: Represents the vertical axis in the 3D space.**Vectors:**1. **Vector V1**: Originates from the origin point (intersection of X, Y, and Z axes) and extends in the direction between the X and L axes.2. **Vector V2**: Also originates from the origin point and extends vertically upwards towards the Z-axis.3. **Vector V3**: Originates from the origin point and extends in the direction closer to the L-axis compared to V1 and V2, showing an intermediate angle between them.**Directional Line (L):**- The line labeled 'L' is positioned in the 3D space and represents the combination of components along X, Y, and Z directions.### Application:This visualization is instrumental in understanding how vectors are defined in a three-dimensional space and how they can be decomposed relative to different axes. It offers groundwork knowledge for subjects involving physics, engineering, computer graphics, and more.For instance, in physics, resolving forces into components along the X, Y, and Z axes helps to analyze the resultant effects of multiple forces acting on a point. In computer graphics, positioning of objects in 3D space involves translating coordinates onto these three axes for rendering 3D models.### Practical Exercise:To further grasp this concept, you can practice by:1. **Identifying components of vectors**: Given a vector, determine its projection onto the X, Y, and Z axes.2. **Creating your own vector diagram**: Using graph paper or modeling software, plot vectors in 3D by combining different axes' components.3. **Solved examples**: Go through detailed examples solving for vector magnitudes and directions using trigonometric relationships and vector properties.

We have to decide there vectors v1, v2, v3 are linearly indipendent in R3 or not.

Answered: In 3-D, let L be a line that does not…

In 3-D, let L be a line that does not pass through the origin. let v1, v2, and v3 be three vector that touch the line. Are the vectors {v1, v2, v3} linearly independent? Or dependent?

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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Question

In 3-D, let L be a line that does not pass through the origin. let v1, v2, and v3 be three vector that touch the line.
Are the vectors {v1, v2, v3} linearly independent? Or dependent?

### Understanding 3D Coordinate Systems and Vector Decomposition

The diagram provided illustrates a 3D coordinate system consisting of three mutually perpendicular axes, labeled X, Y, and Z. In this coordinate system, we have several vectors, each with specific directions and magnitudes.

**Axes:**
1. **X-Axis**: Represents the horizontal axis in the 3D space.
2. **Y-Axis**: Represents the horizontal axis perpendicular to the X-axis in the 3D space.
3. **Z-Axis**: Represents the vertical axis in the 3D space.

**Vectors:**
1. **Vector V1**: Originates from the origin point (intersection of X, Y, and Z axes) and extends in the direction between the X and L axes.
2. **Vector V2**: Also originates from the origin point and extends vertically upwards towards the Z-axis.
3. **Vector V3**: Originates from the origin point and extends in the direction closer to the L-axis compared to V1 and V2, showing an intermediate angle between them.

**Directional Line (L):**
- The line labeled 'L' is positioned in the 3D space and represents the combination of components along X, Y, and Z directions.

### Application:
This visualization is instrumental in understanding how vectors are defined in a three-dimensional space and how they can be decomposed relative to different axes. It offers groundwork knowledge for subjects involving physics, engineering, computer graphics, and more.

For instance, in physics, resolving forces into components along the X, Y, and Z axes helps to analyze the resultant effects of multiple forces acting on a point. In computer graphics, positioning of objects in 3D space involves translating coordinates onto these three axes for rendering 3D models.

### Practical Exercise:
To further grasp this concept, you can practice by:
1. **Identifying components of vectors**: Given a vector, determine its projection onto the X, Y, and Z axes.
2. **Creating your own vector diagram**: Using graph paper or modeling software, plot vectors in 3D by combining different axes' components.
3. **Solved examples**: Go through detailed examples solving for vector magnitudes and directions using trigonometric relationships and vector properties.

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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