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Chapter 11: Problem 80

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

Short Answer

Expert verified
The magnitude of the vector \(\mathbf{v}=\langle 1,0,3\rangle\) is \(\sqrt{10}\).

Step by step solution

01

Identify the components of the vector

The vector \(\mathbf{v}\) has components 1, 0, and 3 in the \(x\), \(y\), and \(z\) directions respectively.
02

Apply the formula for the magnitude of a vector

The magnitude of a vector \(\mathbf{v}=\langle x,y,z\rangle\) is given by \(|\mathbf{v}|=\sqrt{x^2+y^2+z^2}\). Substitute \(x=1\), \(y=0\), and \(z=3\) into the formula.
03

Calculate the magnitude

The magnitude is \(|\mathbf{v}|=\sqrt{1^2+0^2+3^2}=\sqrt{1+9}=\sqrt{10}\)

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Key Concepts

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

Vector Components
In any vector, components are the building blocks that define it in a coordinate system. A vector is denoted by an ordered collection of numbers, representing its direction and length. For example, the vector \(\mathbf{v}=\langle 1,0,3\rangle\) has three components:
  • 1 in the \(x\)-direction
  • 0 in the \(y\)-direction
  • 3 in the \(z\)-direction
Each component tells you how far the vector extends along each axis of the coordinate system. With these numbers, you know not just where the vector starts but how it moves through space.
Magnitude Formula
The magnitude of a vector is essentially its length or size. In the context of a vector \(\mathbf{v}=\langle x,y,z \rangle\), the magnitude is calculated using the following formula:\[|\mathbf{v}|=\sqrt{x^2+y^2+z^2}\]This formula is derived from the Pythagorean theorem, which relates to the lengths of sides in right triangles. Each component of the vector is squared, summed, and the square root of the result is taken. This way, you're calculating the diagonal distance across all dimensions of the vector. For \(\mathbf{v}=\langle 1,0,3 \rangle\), substituting the components into the formula gives:
  • \(x = 1\), \(y = 0\), \(z = 3\)
  • \(\sqrt{1^2 + 0^2 + 3^2} = \sqrt{1 + 0 + 9} = \sqrt{10}\)
This tells us that the vector's magnitude is \(\sqrt{10}\).
Three-Dimensional Vectors
Vectors in three-dimensional space are crucial for describing physical phenomena that occur in the real world. They help in visualizing and solving problems involving forces, velocities, and other vector quantities. A three-dimensional vector like \(\mathbf{v}=\langle 1,0,3 \rangle\) exists in a space with three axes, usually labeled as \(x\), \(y\), and \(z\). When considering these vectors:
  • The component along the \(x\)-axis represents the horizontal direction.
  • The \(y\)-axis component is vertical if you're thinking of an upright plane, but it can vary based on the context.
  • The \(z\)-axis component adds depth, which is crucial for full spatial movement.
This layout allows for accurate modeling of scenarios and helps in making calculations that reflect our tangible reality. Understanding each component's contribution helps to see how vectors interact with the environment around them.

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Most popular questions from this chapter

Find the point(s) of intersection (if any) of the plane and the line. Also determine whether the line lies in the plane.\(2 x-2 y+z=12, \quad x-\frac{1}{2}=\frac{y+(3 / 2)}{-1}=\frac{z+1}{2}\)

Let \(\mathbf{r}=\langle x, y, z\rangle\) and \(\mathbf{r}_{0}=\langle 1,1,1\rangle .\) Describe the set of all points \((x, y, z)\) such that \(\left\|\mathbf{r}-\mathbf{r}_{0}\right\|=2\)

In Exercises 91 and 92 , determine the values of \(c\) that satisfy the equation. Let \(\mathbf{u}=\mathbf{i}+\mathbf{2} \mathbf{j}+\mathbf{3} \mathbf{k}\) and \(\mathbf{v}=\mathbf{2} \mathbf{i}+\mathbf{2} \mathbf{j}-\mathbf{k}\) \(\|c \mathbf{v}\|=5\)

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Find the distance between the point and the line given by the set of parametric equations.\((4,-1,5) ; \quad x=3, \quad y=1+3 t, \quad z=1+t\)
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