A position vector is a mathematical representation of the location of a point in space relative to a reference point or origin. It is a vector quantity that specifies both the direction and the distance from the origin to the point.
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The position vector of a point $\vec{r}$ is denoted as $\vec{r} = (x, y, z)$, where $x$, $y$, and $z$ are the coordinates of the point in a three-dimensional Cartesian coordinate system.
The magnitude of the position vector, $|\vec{r}|$, represents the distance between the origin and the point.
The direction of the position vector is determined by the angles it makes with the positive $x$, $y$, and $z$ axes.
Position vectors can be added or subtracted to find the displacement between two points in space.
The dot product and cross product of position vectors can be used to calculate various geometric properties, such as the distance between two points or the area of a parallelogram formed by two vectors.
Review Questions
Explain how the position vector is used to represent the location of a point in space.
The position vector $\vec{r} = (x, y, z)$ is used to represent the location of a point in a three-dimensional Cartesian coordinate system. The components $x$, $y$, and $z$ specify the coordinates of the point relative to the origin, which is the fixed reference point at (0, 0, 0). The magnitude of the position vector, $|\vec{r}|$, represents the distance from the origin to the point, while the direction of the vector is determined by the angles it makes with the positive $x$, $y$, and $z$ axes.
Describe how position vectors can be used to calculate the displacement between two points in space.
The displacement between two points in space can be calculated using position vectors. If the position vectors of the two points are $\vec{r_1} = (x_1, y_1, z_1)$ and $\vec{r_2} = (x_2, y_2, z_2)$, then the displacement vector $\vec{d}$ is given by the difference between the two position vectors: $\vec{d} = \vec{r_2} - \vec{r_1} = (x_2 - x_1, y_2 - y_1, z_2 - z_1)$. The magnitude of the displacement vector, $|\vec{d}|$, represents the distance between the two points.
Explain how the dot product and cross product of position vectors can be used to analyze the geometric properties of points in space.
The dot product and cross product of position vectors can be used to analyze various geometric properties of points in space. The dot product of two position vectors, $\vec{r_1} \cdot \vec{r_2}$, can be used to calculate the distance between the two points, as well as the angle between the vectors. The cross product of two position vectors, $\vec{r_1} \times \vec{r_2}$, can be used to calculate the area of the parallelogram formed by the two vectors, which is important in applications such as finding the normal vector to a plane. These vector operations allow for a deeper understanding of the geometric relationships between points in three-dimensional space.
A vector is a mathematical quantity that has both magnitude (size or length) and direction. Vectors are used to represent physical quantities such as force, velocity, and displacement.
Displacement: Displacement is the change in position of an object, represented by a vector that points from the initial position to the final position.