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Chapter 3: Problem 7

Use the given equation of a line to find a point on the line and a vector parallel to the line. \mathbf{x}=(1-t)(4,6)+t(-2,0)

Short Answer

Expert verified
Point on the line: (4,6); Parallel vector: (-2,0).

Step by step solution

01

Identifying the Form of the Parametric Equation

The equation \( \mathbf{x} = (1-t)(4,6) + t(-2,0) \) is a parametric equation of a line. Here, \((4,6)\) is one point on the line and \((-2,0)\) functions as the direction vector of the line since it scales directly with \(t\).
02

Finding a Point on the Line

To find a specific point on the line, substitute a specific value of \(t\) into the equation. Setting \(t = 0\), substitutes into the equation and yields: \( \mathbf{x} = (1-0)(4,6) = (4,6) \). Thus, \((4,6)\) is a point on this line.
03

Determining a Parallel Vector

The vector \((-2,0)\) is parallel to the line. This is the vector multiplied by \(t\) in the parametric form \( (1-t)(4,6) + t(-2,0) \). It represents the direction in which the line extends.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vectors
A vector is a mathematical object often represented as an arrow with both direction and magnitude. In the parametric equation given, vectors play a crucial role in defining both a point on the line and the direction of the line itself. Vectors are essential in representing points in space and in determining directions, such as in velocity or force.
  • Vectors are typically written in component form, like \((x, y)\) in two dimensions or \((x, y, z)\) in three dimensions.
  • The direction vector in this exercise is \((-2, 0)\), which indicates the line's direction, and scales with the parameter \(t\).
  • A vector's magnitude can be calculated using the Pythagorean theorem: if a vector is \((a, b)\), its magnitude is \(\sqrt{a^2 + b^2}\).
Understanding vectors is vital for solving physics problems, engineering tasks, and any field that works with directions and speeds.
Linear Algebra
Linear algebra is the branch of mathematics concerning linear equations, linear functions, and their representations through matrices and vector spaces. It proves indispensable in solving systems of linear equations, which are foundational in mathematics and many applied sciences.
  • Linear algebra involves the study of vectors and matrices, which facilitate the representation of linear systems and transformations.
  • Parametric equations, like \(\mathbf{x} = (1-t)(4,6) + t(-2,0)\), are a linear algebra concept used to describe lines in space through vectors.
  • This form allows easy manipulation and understanding of how variables, or in this case, the parameter \(t\), affect the line and its position in space.
With linear algebra, we can solve complex systems, compute transformations, and analyze geometric transformations.
Lines in Space
Lines in space are best expressed through parametric equations, which involve a specific point on the line and a direction vector. These characteristics describe how the line moves through space based on a parameter, usually \(t\).
  • Any line can be defined using a point and a direction vector. Here, the line uses \((4, 6)\) as the point and \((-2, 0)\) as the direction vector.
  • The parametric form, \((1-t)(4,6) + t(-2,0)\), allows us to easily find any point on the line by modifying the parameter \(t\).
  • A fixed \(t\) gives a specific point, way to determine the entirety of the line by varying \(t\).
Lines in space are integral to geometry, CAD applications, and any field requiring spatial reasoning.

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Most popular questions from this chapter

Find the distance between the given parallel planes. $$2 x-y-z=5 \text { and }-4 x+2 y+2 z=12$$

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