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Chapter 14: Problem 4

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

Short Answer

Expert verified
The position vectors are \(\vec{r_1} = 9i + 1j - 1k\) and \(\vec{r_2} = -4i + 0j + 4k\).

Step by step solution

01

Understanding Position Vectors

Position vectors are used to represent points in a coordinate system relative to the origin. Given a 3-dimensional point \(x, y, z\), its position vector is written in the form of \(\vec{r} = xi + yj + zk\), where \(i, j, k\) are the unit vectors along the x, y, and z axes respectively.
02

Finding Position Vector for Point (9, 1, -1)

For the point with coordinates \(9, 1, -1\), the position vector is formed using its components as coefficients of the unit vectors. Thus, the position vector is \(\vec{r_1} = 9i + 1j - 1k\).
03

Finding Position Vector for Point (-4, 0, 4)

For the point with coordinates \(-4, 0, 4\), similarly, the position vector is formed. It will be \(\vec{r_2} = -4i + 0j + 4k\).

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Key Concepts

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

3-dimensional coordinates
When discussing points in space, we often refer to their 3-dimensional coordinates. These coordinates are expressed as \(x, y, z\) and help us locate points precisely in a 3-dimensional space, much like giving a street address helps locate a house in a city. Each coordinate corresponds to the position along the respective x, y, or z axis in a Cartesian coordinate system.
  • The x-coordinate tells us how far along the x-axis (left-right) the point is.
  • The y-coordinate indicates the location along the y-axis (up-down).
  • The z-coordinate represents the depth or motion along the z-axis (in-out).
By combining these three coordinates, we can pinpoint any location in a 3-dimensional space. This system is fundamental in mathematics, physics, computer graphics, and many technological fields that require precise spatial representation.
vector representation
Vectors are incredible tools in mathematics and physics, offering a way to depict both magnitude and direction. A vector in 3-dimensional space, like a position vector, is typically represented in terms of its components along the x, y, and z axes. For a vector pointing from the origin to the point \(x, y, z\), the vector is described as \(\vec{r} = xi + yj + zk\).
  • Here, \(i, j, k\) are unit vectors that align with the x, y, and z axes respectively.
  • The coefficients \(x, y, z\) provide the scale or size of the vector in each direction.
This representation allows us to easily perform operations such as addition, subtraction, and scalar multiplication of vectors, which are widely used in physics to calculate forces, velocities, and other vector quantities.
unit vectors
Unit vectors play a crucial role in the vector representation by defining direction without concerning magnitude. They have a magnitude of one and inform us precisely which direction a particular axis is pointing. In the context of 3-dimensional coordinates, we use the unit vectors: \(i, j, k\).
  • The unit vector \(i\) points in the direction of the x-axis.
  • The unit vector \(j\) points in the direction of the y-axis.
  • The unit vector \(k\) points in the direction of the z-axis.
Using these unit vectors, we can effectively build any other vector by scaling them. For example, the vector \(9i + 1j - 1k\) is a combination of these unit vectors, showing us exactly how far to travel along each axis from the origin.
coordinate system
To discuss points and vectors in space, it's essential to understand the coordinate system, particularly the Cartesian system used in 3-dimensional space. It consists of three perpendicular axes: the x, y, and z axes, intersecting at a point known as the origin.
  • The origin is the reference point, usually denoted as \((0, 0, 0)\).
  • The x-axis runs horizontally; the y-axis runs vertically.
  • The z-axis adds depth, making the system 3-dimensional.
Each point in this space is specified by its distance from the origin along these axes. This right-handed coordinate system is an indispensable framework for navigating space and understanding spatial relationships in various applications, such as engineering and physics.

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Most popular questions from this chapter

If A has coordinates \((-4,2,1)\) and B has coordinates \((2,0,2)\) find the direction ratio of the vector \(\overrightarrow{\mathrm{AB}}\). Find its direction cosines \(l, m\) and \(n\) and verify that \(l^{2}+m^{2}+n^{2}=1\).

Write down the vector equation of the line passing through the points with position vectors $$ \begin{aligned} &p=3 i+7 j-2 k \\ &q=-3 i+2 j+2 k \end{aligned} $$ Find also the cartesian equation of this line.

Two vectors have moduli 7 and 13 respectively. The angle between them is \(45^{\circ}\). Evaluate their scalar product.

Find the equation of the plane with normal \(j\) that is a distance 2 from the origin.

If the triple scalar product \((\boldsymbol{a} \times \boldsymbol{b}) \cdot c\) is equal to zero, then (i) \(\boldsymbol{a}=\mathbf{0}\), or \(\boldsymbol{b}=\mathbf{0}\), or \(\boldsymbol{c}=\mathbf{0}\) or (ii) two of the vectors are parallel, or (iii) the three vectors lie in the same plane (they are said to be coplanar).
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