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Chapter 14: Problem 1

Explain the distinction between a position vector and a more general or free vector.

Short Answer

Expert verified
Answer: A position vector represents the position of a point or object in a coordinate system with respect to a reference point, usually the origin, while a free vector has magnitude and direction but is not fixed to any specific location. Position vectors are used to specify the locations of points or objects, while free vectors are used to describe quantities such as forces, velocities, and accelerations that are independent of the location in the coordinate system.

Step by step solution

01

1. Define position vector

A position vector is a vector that represents the position of a point or object in a coordinate system with respect to a reference point, usually the origin. It has both magnitude (length) and direction, pointing from the origin to the location of the point. In a cartesian coordinate system, the position vector is usually denoted as \(\textbf{r}=(x,y,z)\), where x, y, and z are the coordinates of the point.
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2. Define free or general vector

A free or general vector, also known as "bound vector," is a vector that has magnitude and direction but is not fixed to any specific location. It can be translated anywhere within the coordinate system without changing its magnitude or direction. Essentially, it represents a displacement rather than a position. Examples of free vectors include force, velocity, and acceleration vectors.
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3. Differences in representation

A position vector is usually represented with an arrow originating from the origin of the coordinate system and ending at the position of the point, while a free vector can start and end anywhere in the coordinate system, as long as its magnitude and direction remain the same. The position vector maintains a connection with the origin, unlike the free vector, which can be moved freely.
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4. Differences in applications

Position vectors and free vectors have different applications in various fields of science and engineering. Position vectors are used to specify the locations of points or objects in the coordinate system, while free vectors are used to describe quantities such as forces, velocities, and accelerations that are independent of the location in the coordinate system. In essence, position vectors represent location, while general vectors represent quantities that can be applied to any point in space.
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5. Addition and subtraction

For position vectors, subtraction is used to find the difference between two positions, resulting in a displacement vector or a free vector. For example, given two position vectors \(\textbf{r}_1\) and \(\textbf{r}_2\), the free vector \(\textbf{d}_{12} = \textbf{r}_2 - \textbf{r}_1\) represents the displacement from point 1 to point 2. Adding or subtracting free vectors, on the other hand, often represents the sum or difference of quantities such as forces, velocities, and accelerations.

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Most popular questions from this chapter

State the coordinates of the point \(\mathrm{P}\) if its position vector is given as (a) \(3 \boldsymbol{i}-7 \boldsymbol{j}\), (b) \(-4 \boldsymbol{i}\), (c) \(-0.5 \boldsymbol{i}+13 \boldsymbol{j}\), (d) \(a \boldsymbol{i}+b \boldsymbol{j}\).

Find the modulus of the vector $$ p=2 i-j+5 k $$

On a diagram show the arbitrary vectors \(p\) and q. Then show the following: (a) \(p+q\) (d) \(4 q\) (e) \(-2 q\) (c) \(q-p\).

State the position vectors of the points with coordinates \((9,1,-1)\) and \((-4,0,4)\).

Given three vectors \(\boldsymbol{a}, \boldsymbol{b}\) and \(\boldsymbol{c}\), their triple scalar product is defined to be \((\boldsymbol{a} \times \boldsymbol{b}) \cdot \boldsymbol{c}\). It can be shown that the modulus of this is the volume of the parallelepiped formed by the three vectors. Find the volume of the parallelepiped formed by the three vectors \(\boldsymbol{a}=3 \boldsymbol{i}+\boldsymbol{j}-2 \boldsymbol{k}, \boldsymbol{b}=\boldsymbol{i}+2 \boldsymbol{j}-2 \boldsymbol{k}\) and \(\boldsymbol{c}=2 \boldsymbol{i}+5 \boldsymbol{j}+\boldsymbol{k}\)
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