Question

Find the magnitude of \(v\). \(\mathbf{v}=\langle 1,0,3\rangle\)

Short answer

The magnitude of the vector \(\mathbf{v}\) is \(\sqrt{10}\)

Step-by-step solution

Identify the components of the vector

The vector \(\mathbf{v}\) is given by \(1,0,3\), so the components are \(v_1 = 1\), \(v_2 = 0\), and \(v_3 = 3\).

Square each component

The square of each component is then calculated as \(v_1^2 = 1^2 = 1\), \(v_2^2 = 0^2 = 0\), and \(v_3^2 = 3^2 = 9\).

Sum the squared components

Adding these squares together gives the sum \(1 + 0 + 9 = 10\).

Take the square root of the sum

Finally, taking the square root of this sum gives the magnitude of vector \(\mathbf{v}\), that is \(\|v\| = \sqrt{10}\).

Key concepts

Vector Components

When working with vectors, especially in physics and engineering, understanding vector components is crucial. A vector can be thought of as an arrow that has both direction and magnitude. In three-dimensional space, any vector can be broken down into three components, each representing the vector's projection onto the x, y, and z axes respectively.

These components are often written as \(\langle v_1, v_2, v_3 \rangle\), correlating to the vector’s horizontal (x), vertical (y), and depth (z) influences. For instance, in the exercise given \(\mathbf{v}=\langle 1,0,3\rangle\), the vector \(\mathbf{v}\) is composed of three components: \(v_1 = 1\) along the x-axis, \(v_2 = 0\) along the y-axis, and \(v_3 = 3\) along the z-axis. The component \(v_2 = 0\) implies that there is no movement in the y-direction.

Understanding each component is fundamental since each influences the vector's direction and magnitude. The computation of the vector's magnitude, indeed, relies heavily on these individual components.

Squaring Vector Components

When calculating the magnitude of a vector, a critical step involves squaring its components. Squaring, in mathematical terms, means multiplying a number by itself. For vector components, this action helps us find the sum of the squares, which is used in calculating the vector's length or magnitude using the Pythagorean theorem.

Let's consider the vector from the exercise, \(\mathbf{v}=\langle 1,0,3\rangle\). To square its components, you would calculate \(v_1^2 = 1^2 = 1\), \(v_2^2 = 0^2 = 0\), and \(v_3^2 = 3^2 = 9\). This step transforms the vector components into a form that can be used within the Pythagorean theorem to find the magnitude of the vector.

It is important to highlight that squaring a component also results in a positive value, irrespective of the original component's sign. Therefore, it contributes positively to the magnitude of the vector, reinforcing the idea that vector magnitude is never negative.

Pythagorean Theorem in Three Dimensions

The Pythagorean theorem is a well-known formula used to calculate the length of the hypotenuse of a right triangle. However, in three-dimensional space, we extend this concept to find the magnitude of a vector. This three-dimensional extension involves squaring each vector component, summing these squares, and then taking the square root of this sum to find the vector's magnitude.

For a three-dimensional vector \(\mathbf{v}=\langle v_1, v_2, v_3 \rangle\), the magnitude (denoted as \(\|\mathbf{v}\|\)) is given by \(\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2}\). In the exercise, summing the squared components \(1^2 + 0^2 + 3^2\) gives us 10. Taking the square root of 10 yields the magnitude \(\|\mathbf{v}\| = \sqrt{10}\).

This three-dimensional approach to the Pythagorean theorem is a fundamental concept for calculating vector magnitudes in space. It allows for the determination of the 'length' of the vector regardless of direction, as illustrated in the given exercise.