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

See similar textbooks

Related questions

Q: Find vectors u and v whose initial and terminal points are given. Determine whether u and v are…

A: The u vector can be computed as: u=6-1i+11-1ju=5i+10j

Q: 2 Write the vector x= as a linear combination of the vectors -1 1) u=0 |, v=| 1 w= 1 1 N O H O O

A: Consider x = au+bv+cw then 20-1 = a100+b110+c11120-1 = a+b+cb+cc On comparing the coefficients , we…

Q: Find the vector equation of the line that goes through the point (10,15,9)(10,15,9) and is…

Q: Write the vector equation of the line defined by the following points. (25,17,42)and(-16, 12, 25)

A: Answer is given below :-Explanation:To find the vector equation of the line passing through two…

Q: Find the vector v = PQ. P = (9,4,7) Q=(10, 7, 3)

Q: Which of the following represents vector u = (37,-51) in linear form?

A: u=-51i+37ju=37i+51ju=37i-51ju=51i-37j

Q: has an initial point (-3, -6) and terminal point (4, 5). Write as a linear combination of the…

Q: et V₁ = (1, -2,-2), V₂ = (0, -1, -2), V3 = (2, -4,-5). re the vectors 1, 2 and 3 linearly…

Q: Given, vector a = (3, 4, -2), vector b = (2, -7, 1), and vector c = (-6, 5, 4). Evaluate |-2b x c|

Q: Determine whether the vectors v₁ =(4,-8, 12), and v₂ = (-5, 10, 15) are linearly independent; if…

Q: The zero vector 0 = (0, 0, 0) can be written as a linear combination of the vectors V₁, V₂, and v₂…

A: Linear Combination: A vector V is called a linear combination of vectors v1, v2, . . ., vn…

Q: Find the vector

A: The vector r=<a,b> is written as, r(i)=ai+ bj

Q: Determine whether the vectors v₁ = (1, 2), v₂ = (-1,0) and v3 = (0, 1) are linearly independent; if…

Q: Suppose that v₁ = (4, 4, 0, 7), v₂ = (2, -5, 5, 4) and v3 = (-7, 0, 5, 2). Is the following vector…

Q: Find two linearly independent vectors perpendicular to the vector 2 -5 -9 D =

A: Two vectors are said to be perpendicular to each other if and only if their dot product is zero.

Q: Determine a so that the vector u = {6,3} is a linear combination u = av+bw of vectors v = {-3, 3}, w…

Q: Determine whether the two given vectors are orthogonal, parallel, or neither. u = (9, 3), v = (-2,…

A: The given two vectors are u=9, 3 and v=-2, 6 We have to determine whether the given two vectors are…

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.

Expert Solution

This question has been solved!

Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.

This is a popular solution!

SEE SOLUTION Check out a sample Q&A here

Our aim :

Main concept :

Explanation

Answer :

Trending now

This is a popular solution!

Step by step

Solved in 4 steps

SEE SOLUTION Check out a sample Q&A here

Knowledge Booster

$Vector Addition and Scalar Multiplication$

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.

Similar questions

Find the vector equation of the line that contains the points (-5, 17,9) and (-17,10,2). r(t) = ( ) +t ( 9 "
Consider the parallelogram with vertices A(-3, 0), B(-1, 3), C(5, 2), and D(3, −1). Find AB and AD, writing each as a three-dimensional vector. AB = (2,3,0) AD = (6, −1, − 1) Find AB × AD. AB × AD = (0,0,9) Find the area of the parallelogram. 9
Determine if the vector b 5 can be written as a linear combination of the vectors V1 = 1 and v₂ = 2 True False = 2 3 LO 2
Determine if the pairs of vectors below are "parallel", "orthogonal", or "neither". a = (-5,5,-5) and b = (-10, 10, 20) are neither a = (-5,5,-5) and b = (-10, 10, -10) are neither a = (-5,5,-5) and b = (25, -25, 25) are neither
Let Are the vectors 7₁, 72 and 3 linearly independent? linearly dependent If the vectors are independent, enter zero in every answer blank since those are only the values that make the equation below true. If they are dependent, find numbers, not all zero, that make the equation below true. You should be able to explain and justify your answer. -O]++] 1 26 V₁ = = [3₁]. * = [9] = [²6]- ₁ V3 26
Determine if the vectors vi = (1, –1,3, –1), v2 = (1, -1,4, 2), v3 = (1, -1, 5, 7) are linearly dependent or linearly independent.
The zero vector 0 = (0, 0, 0) can be written a linear combination of the vectors v., v,, and v- because 0 = Ov, + 0v, + 0v3. This is called the trivia! solution. Can you find a nontrivial way of writing 0 as a linear combination of the three vectors? (Enter your answer in terms of v., v,, and v- If not possible, enter IMPOSSIBLE.) V1 = (1, 0, 1), V2 = (-1, 1, 2), V3 = (0, 1, 6)
Give the vector equation of the line passing through P and Q. P= (3, 1), Q = (2, 0) X = + t
If possible express the vector v as a linear combination of V₁, V2, and V3.
Determine whether the vector U = 2 is a linear combination of the vectors X = 2 and X3 =-1? Give reason(s) for your response.
explain briefly ...
Find the vector equation of the line passing through the point (1, 2, - 4) and x-15 and 7 X-8 y+19 z-10 y-29 z-5 -16 3 8 -5 perpendicular to the two lines: