Question

Writing The initial and terminal points of the vector \(\mathbf{v}\) are \(\left(x_{1}, y_{1}, z_{1}\right)\) and \((x, y, z) .\) Describe the set of all points \((x, y, z)\) such that \(\|\mathbf{v}\|=4\)

Short answer

The set of points \((x, y, z)\) such that \(\|\mathbf{v}\|=4\) describes a sphere with center \((x_1, y_1, z_1)\) and radius 4.

Step-by-step solution

Find the vector \(\mathbf{v}\)

The vector \(\mathbf{v}\) can be represented as \(\mathbf{v} = (x - x_1, y - y_1, z - z_1)\).

Formulate the magnitude equation

The magnitude of a vector can be found using the equation \(\|\mathbf{v}\| = \sqrt{(x - x_1)^2 + (y - y_1)^2 + (z - z_1)^2 }\). Since the magnitude of vector \(\mathbf{v}\) is given to be 4, we can set up the equation \(\sqrt{(x - x_1)^2 + (y - y_1)^2 + (z - z_1)^2 } = 4\).

Simplify the equation

Square both sides of the equation to get rid of the square root in order to simplify the equation. The equation becomes \((x - x_1)^2 + (y - y_1)^2 + (z - z_1)^2 = 16\)

Description of the set of all points

The equation that we have derived is the equation of a sphere with center at \((x_1, y_1, z_1)\) and radius 4. Thus, the set of all points \((x, y, z)\) is the set of all points on this sphere.

Key concepts

Magnitude of a Vector

The magnitude of a vector is a measure of the length or size of the vector. In the world of physics and mathematics, it is often essential to determine how far a point has traveled from its starting position, which, in vector terms, is the magnitude. For a three-dimensional vector given by its terminal point \( (x, y, z) \) compared to its initial point \( (x_{1}, y_{1}, z_{1}) \) such as the vector \( \mathbf{v} = (x - x_1, y - y_1, z - z_1) \), the magnitude is calculated using the Euclidean distance formula: \( \|\mathbf{v}\| = \sqrt{(x - x_1)^2 + (y - y_1)^2 + (z - z_1)^2 } \).

This formula is rooted in the Pythagorean theorem and can be thought of as a way to measure the 'straight-line' distance between two points in three-dimensional space. When the exercise asks to describe the set of all points \( (x, y, z) \) such that \( \|\mathbf{v}\| = 4 \) it suggests that we are looking for all the points that are a fixed distance away from a certain point – which shapes a sphere, as discovered in the problem-solving process.

Euclidean Vector

A Euclidean vector, often simply called a vector, represents quantity having both direction and magnitude. In a Euclidean space, it can be pictorially represented as an arrow: the direction of the arrow corresponds to the direction of the vector, while the length of the arrow is proportional to the vector's magnitude.

Euclidean vectors are essential in various branches of science and engineering, as they can describe velocity, acceleration, and other physical quantities. They are not fixed in position and can be moved parallel to themselves in a space, as long as the magnitude and direction remain unchanged. When it comes to calculations involving vectors in three-dimensional space, operators like addition, subtraction, and even multiplication have specific geometrical interpretations.

Vector Algebra

Delving into the realm of vector algebra, one encounters operations that can be applied to vectors such as addition, subtraction, and scalar multiplication. These operations follow specific rules that differ from ordinary algebra.

For example, to add two vectors, you 'place' the initial point of one vector at the terminal point of another and then draw a vector from the free initial point to the free terminal point; this is known as the triangle rule. Another rule to remember is that when you multiply a vector by a scalar, you alter the magnitude of the vector but not its direction.

Scalar Multiplication

In the context of the given problem, no vector multiplication is involved, but understanding how to manipulate the magnitude of vectors can help elucidate the concept of a sphere equation derived from the condition set on the magnitude of the vector.

Sphere Equation

The sphere equation is a mathematical description of the set of all points in space that are at a fixed distance (the radius) from a central point (the center). The general form of a sphere's equation in a three-dimensional Cartesian coordinate system is \( (x - h)^2 + (y - k)^2 + (z - l)^2 = r^2 \), where \( (h, k, l) \) is the center of the sphere and \( r \) is the radius.

In the problem provided, the sphere equation emanates from setting the magnitude of the vector equal to 4. This leads to the equation \( (x - x_1)^2 + (y - y_1)^2 + (z - z_1)^2 = 16 \), where \( 16 \) is the square of \( 4 \), the radius of the sphere, and the point \( (x_1, y_1, z_1) \) serves as the center. The sphere equation represents the locus of points that satisfy the condition of being a constant distance from a central point, contributing to the visualization and understanding of spatial geometry.