Elementary linear algebra questions

In summary, elementary linear algebra is a branch of mathematics that deals with the study of linear equations and their properties, involving the analysis and manipulation of vectors and matrices. The basic concepts of elementary linear algebra include vectors, matrices, linear transformations, systems of linear equations, determinants, eigenvalues and eigenvectors, and vector spaces. It has many practical applications in fields such as computer graphics, data analysis, robotics, and economics, as well as in solving problems related to forces, motion, and electrical circuits in engineering and physics. Key properties of elementary linear algebra include commutative, associative, and distributive properties, as well as the properties of inverse and identity matrices. Common methods for solving elementary linear algebra problems include Gaussian elimination, matrix
  • #1
tigerseye
16
0
Could someone help me please?

1.) Find the scalar equation of the line containing P(2, -1, 3) and perpendicular to the lines [x y z]^T=[4 -1 2]^T + t[7 0 1] and [x y z]^T=[-2 0 1] + t[2 3 0]^T.

2.)Find all points C on the line through A(1, -1, 2) and B(2, 0, 1) such that vectors llACll= 2 llBCll.
 
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  • #2
as they say here: what have you tried?
 
  • #3


Sure, I can help you with these questions. Let's start with the first one.

1.) To find the scalar equation of the line perpendicular to the given lines, we first need to find the direction vector of the line. This can be done by taking the cross product of the direction vectors of the two given lines. So, the direction vector of the perpendicular line will be:

d = [7 0 1] x [2 3 0] = [3 -7 14]

Now, we can use the point-slope form of a line to find the equation. The point P(2, -1, 3) lies on the line, so we can use it as our point. The point-slope form is given by:

(x-x1)/a = (y-y1)/b = (z-z1)/c

where (a, b, c) is the direction vector and (x1, y1, z1) is the given point. Plugging in the values, we get:

(x-2)/3 = (y+1)/(-7) = (z-3)/14

This is the scalar equation of the line containing P(2, -1, 3) and perpendicular to the given lines.

2.) To find the points C on the line through A(1, -1, 2) and B(2, 0, 1) such that vectors llACll= 2 llBCll, we can use the distance formula between two points. The distance formula is given by:

d = √((x2-x1)^2 + (y2-y1)^2 + (z2-z1)^2)

where (x1, y1, z1) and (x2, y2, z2) are the two points. We can set up two equations using this formula, one for the distance between A and C and one for the distance between B and C. Since we want the distances to be equal, we can set the two equations equal to each other:

√((x-1)^2 + (y+1)^2 + (z-2)^2) = 2√((x-2)^2 + y^2 + (z-1)^2)

Simplifying and squaring both sides, we get:

(x-1)^2 + (y+1)^2 + (z-
 

Related to Elementary linear algebra questions

1. What is elementary linear algebra?

Elementary linear algebra is a branch of mathematics that deals with the study of linear equations and their properties. It involves the analysis and manipulation of vectors and matrices to solve problems in various fields such as engineering, physics, and computer science.

2. What are the basic concepts of elementary linear algebra?

The basic concepts of elementary linear algebra include vectors, matrices, linear transformations, systems of linear equations, determinants, eigenvalues and eigenvectors, and vector spaces.

3. How is elementary linear algebra used in real life?

Elementary linear algebra has many practical applications in real life, such as in computer graphics, data analysis, robotics, and economics. It is also used in fields like engineering and physics to solve problems related to forces, motion, and electrical circuits.

4. What are the key properties of elementary linear algebra?

Some key properties of elementary linear algebra include the commutative, associative, and distributive properties, as well as the properties of inverse and identity matrices. These properties are essential for performing operations on matrices and solving systems of linear equations.

5. What are some common methods for solving elementary linear algebra problems?

There are several methods for solving elementary linear algebra problems, including Gaussian elimination, matrix operations, and determinants. Other methods include using eigenvalues and eigenvectors, vector projections, and least squares approximations. The method used depends on the specific problem and its requirements.

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