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Chapter 14: Problem 14

Sketch position vectors for \(a, b, 2 a,-3 b\) \(a+b,\) and \(a-b\). $$ \mathbf{a}=-\mathbf{i}+2 \mathbf{j}+3 \mathbf{k}, \quad \mathbf{b}=-2 \mathbf{j}+\mathbf{k} $$

Short Answer

Expert verified
Draw vectors from origin to respective points: \((-1,2,3), (0,-2,1), (-2,4,6), (0,6,-3), (-1,0,3), (-1,4,2)\).

Step by step solution

01

Understand Position Vectors

Position vectors describe the location of a point in space relative to an origin. Here, we are given two vectors \( \mathbf{a} = -\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \) and \( \mathbf{b} = -2\mathbf{j} + \mathbf{k} \). We can represent these vectors starting at the origin and extending to the coordinate specified by each vector.
02

Sketch Vector a

Vector \( \mathbf{a} = -\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \) starts at the origin (0,0,0) and extends to (-1, 2, 3) in 3D space. Draw this vector by marking the point (-1, 2, 3) and connecting it to the origin.
03

Sketch Vector b

Vector \( \mathbf{b} = -2\mathbf{j} + \mathbf{k} \) starts at the origin (0,0,0) and goes to (0, -2, 1) in 3D space. Draw this by locating the endpoint (0, -2, 1) and connecting it to the origin.
04

Calculate and Sketch 2a

Compute \( 2 \mathbf{a} = 2(-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}) = -2\mathbf{i} + 4\mathbf{j} + 6\mathbf{k} \). This vector extends from the origin to (-2, 4, 6). Sketch this vector by extending it to the calculated point.
05

Calculate and Sketch -3b

Compute \( -3\mathbf{b} = -3(-2\mathbf{j} + \mathbf{k}) = 6\mathbf{j} - 3\mathbf{k} \). This vector extends from the origin to (0, 6, -3). Mark this point and draw the vector from the origin to there.
06

Calculate and Sketch a+b

Add vectors \( \mathbf{a} + \mathbf{b} \) to get \( (-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}) + (-2\mathbf{j} + \mathbf{k}) = -\mathbf{i} + 3\mathbf{k} \). This vector extends to (-1, 0, 3). Sketch it from the origin moving to (-1, 0, 3).
07

Calculate and Sketch a-b

Subtract vector \( \mathbf{b} \) from \( \mathbf{a} \) to get \( (-\mathbf{i} + 2\mathbf{j} + 3\mathbf{k}) - (-2\mathbf{j} + \mathbf{k}) = -\mathbf{i} + 4\mathbf{j} + 2\mathbf{k} \). It extends to (-1, 4, 2). Draw it from the origin to this point in the 3D coordinate system.

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Key Concepts

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

Vector Addition
Vector addition involves combining two or more vectors to form a resultant vector. This is done by adding the corresponding components of the vectors together. In our exercise, we needed to find \( \mathbf{a} + \mathbf{b} \).
  • Start with vector \( \mathbf{a} = -\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \).
  • Add vector \( \mathbf{b} = -2\mathbf{j} + \mathbf{k} \).
Combine the components:
  • The \( \mathbf{i} \) component: \( -1 + 0 = -1 \).
  • The \( \mathbf{j} \) component: \( 2 + (-2) = 0 \).
  • The \( \mathbf{k} \) component: \( 3 + 1 = 4 \).
Thus, the vector \( \mathbf{a} + \mathbf{b} = -\mathbf{i} + 4\mathbf{k} \).
Visualize this in a 3D coordinate system by plotting the vector from the origin to the point \((-1, 0, 4)\).
Vector Subtraction
Subtraction of vectors is quite similar to addition, except we're finding the difference between the vectors. We are given \( \mathbf{a} \) and \( \mathbf{b} \) and need to find \( \mathbf{a} - \mathbf{b} \).
  • Vector \( \mathbf{a} = -\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \).
  • Subtract \( \mathbf{b} = -2\mathbf{j} + \mathbf{k} \).
Compute the vector difference:
  • The \( \mathbf{i} \) component: \( -1 - 0 = -1 \).
  • The \( \mathbf{j} \) component: \( 2 - (-2) = 4 \).
  • The \( \mathbf{k} \) component: \( 3 - 1 = 2 \).
Therefore, the new vector \( \mathbf{a} - \mathbf{b} = -\mathbf{i} + 4\mathbf{j} + 2\mathbf{k} \). This vector can be sketched from the origin to the point \((-1, 4, 2)\) in the 3D space.
3D Coordinate System
Understanding the 3D coordinate system is crucial for working with vectors. It consists of three axes: \( x \), \( y \), and \( z \), represented by \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) respectively. Each point in this system is defined by three coordinates \( (x, y, z) \). Here's how you can visualize vectors in a 3D coordinate system:
  • Origin: The point where all axes intersect, typically denoted as \((0, 0, 0)\).
  • Components: Each dimension contributes to defining the position of a vector.
  • The vector \( \mathbf{a} = -\mathbf{i} + 2\mathbf{j} + 3\mathbf{k} \) goes from the origin to \((-1, 2, 3)\).
This approach helps in easily plotting vectors and performing operations like addition and subtraction. Graphically, these vectors can be represented as arrows in the 3D space moving from the origin to their respective coordinates.

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Exer. \(39-50:\) Sketch the graph of the equation. $$ z=9-x^{2}-y^{2} $$

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