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Chapter 9: Problem 24

Express the given vector in terms of the unit vectors i, \(\mathbf{j}\). and \(\mathbf{k}\). $$\mathbf{u}=\langle 3,1,0\rangle, \quad \mathbf{v}=\langle 3,0,-5\rangle$$

Short Answer

Expert verified
\( \mathbf{u} = 3\mathbf{i} + 1\mathbf{j} + 0\mathbf{k} \), \( \mathbf{v} = 3\mathbf{i} - 5\mathbf{k} \).

Step by step solution

01

Understand the Unit Vectors

The unit vectors \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) are the standard basis vectors in three-dimensional space. Specifically, \( \mathbf{i} = \langle 1, 0, 0 \rangle \), \( \mathbf{j} = \langle 0, 1, 0 \rangle \), and \( \mathbf{k} = \langle 0, 0, 1 \rangle \). These vectors represent directions along the x-axis, y-axis, and z-axis, respectively.
02

Identify Components of Vector \( \mathbf{u} \)

The vector \( \mathbf{u} = \langle 3, 1, 0 \rangle \) can be broken down into its components: the x-component is 3, the y-component is 1, and the z-component is 0.
03

Express \( \mathbf{u} \) in Terms of Unit Vectors

Using the components identified, express \( \mathbf{u} \) in terms of unit vectors: \( \mathbf{u} = 3\mathbf{i} + 1\mathbf{j} + 0\mathbf{k} \).
04

Identify Components of Vector \( \mathbf{v} \)

The vector \( \mathbf{v} = \langle 3, 0, -5 \rangle \) can be broken down into its components: the x-component is 3, the y-component is 0, and the z-component is -5.
05

Express \( \mathbf{v} \) in Terms of Unit Vectors

Using the components identified, express \( \mathbf{v} \) in terms of unit vectors: \( \mathbf{v} = 3\mathbf{i} + 0\mathbf{j} - 5\mathbf{k} \).

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Key Concepts

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

Unit Vectors
Unit vectors are fundamental in understanding and working with vectors in any space. They provide a standard way to reference directions without concerning magnitude. In three-dimensional space, the commonly used unit vectors are \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \).

These unit vectors are defined as follows:
  • \( \mathbf{i} = \langle 1, 0, 0 \rangle \) - corresponds to the x-axis direction.
  • \( \mathbf{j} = \langle 0, 1, 0 \rangle \) - corresponds to the y-axis direction.
  • \( \mathbf{k} = \langle 0, 0, 1 \rangle \) - corresponds to the z-axis direction.

These vectors have a magnitude of 1, which is why they are called unit vectors. When expressing a vector in terms of these unit vectors, we essentially decompose the vector into its components along each axis.
Vector Components
Vector components are essentially the building blocks that define a vector in any space. For a three-dimensional vector like \( \mathbf{u} = \langle 3, 1, 0 \rangle \), the components represent how far the vector stretches along each of the respective axes: x, y, and z.

  • The x-component of \( \mathbf{u} \) is 3, indicating it moves three units in the direction of the x-axis.
  • The y-component is 1, showing a single unit movement along the y-axis.
  • The z-component is 0, suggesting that there is no movement in the z-direction.

By expressing a vector such as \( \mathbf{u} \) as \( 3\mathbf{i} + 1\mathbf{j} + 0\mathbf{k} \), we unravel it into its components, making it far more manageable to visualize and use in calculations.
Basis Vectors
Basis vectors form the foundation for a vector space, providing a frame of reference where any vector can be expressed. In a three-dimensional space, the set of unit vectors \( \{\mathbf{i}, \mathbf{j}, \mathbf{k}\} \) can act as basis vectors.

When you express a vector like \( \mathbf{v} = \langle 3, 0, -5 \rangle \) in terms of these basis vectors:
  • \( \mathbf{v} = 3\mathbf{i} \) means 3 units in the x-direction.
  • \( + 0\mathbf{j} \) implies no movement in the y-direction.
  • \( - 5\mathbf{k} \) depicts a movement of 5 units in the opposite z direction.

This representation allows vectors to be expressed in a consistent, standardized way. This is crucial for operations like vector addition, subtraction, and scalar multiplication, facilitating easy manipulation and calculation in vector algebra.

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Most popular questions from this chapter

Three vectors \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\) are given. \(\mathbf{(a)}\) Find their scalar triple product \(\mathbf{a} \cdot(\mathbf{b} \times \mathbf{c}) .\) \(\mathbf{(b)}\) Are the vectors coplanar? If not, find the volume of the parallelepiped that they determine. $$\mathbf{a}=\langle 3,0,-4\rangle, \quad \mathbf{b}=\langle 1,1,1\rangle, \quad \mathbf{c}=\langle 7,4,0\rangle$$

Find the work done by the force \(\mathbf{F}\) in moving an object from \(P\) to \(Q\). $$\mathbf{F}=10 \mathbf{i}+3 \mathbf{j} ; \quad P(2,3), Q(6,-2)$$

Let a \(=\langle 2,2,2\rangle\) \(\mathbf{b}=\langle- 2,-2,0\rangle,\) and \(\mathbf{r}=\langle x, y, z\rangle\) (a) Show that the vector equation \((\mathbf{r}-\mathbf{a}) \cdot(\mathbf{r}-\mathbf{b})=0\) represents a sphere, by expanding the dot product and simplifying the resulting algebraic equation. (b) Find the center and radius of the sphere. (c) Interpret the result of part (a) geometrically, using the fact that the dot product of two vectors is 0 only if the vectors are perpendicular. endpoints of the vectors \(\mathbf{a}, \mathbf{b},\) and \(\mathbf{r},\) noting that the end. points of a and b are the endpoints of a diameter and the endpoint of \(\mathbf{r} \text { is an arbitrary point on the sphere. }]\) (d) Using your observations from part (a), find a vector equation for the sphere in which the points \((0,1,3)\) and \((2,-1,4)\) form the endpoints of a diameter. Simplify the vector equation to obtain an algebraic equation for the sphere. What are its center and radius?

Find the horizontal and vertical components of the vector with given length and direction, and write the vector in terms of the vectors i and j. $$|\mathbf{v}|=1, \quad \theta=225^{\circ}$$

Find the magnitude and direction (in degrees) of the vector. $$\mathbf{v}=\langle 40,9\rangle$$
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