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Chapter 13: Problem 23

Perform the indicated operations on the following vectors: $$\begin{aligned} &\vec{a}=2 \vec{j}+\vec{k}, \quad \vec{b}=-3 \vec{i}+5 \vec{j}+4 \vec{k}, \quad \vec{c}=\vec{i}+6 \vec{j},\\\ &\vec{x}=-2 \vec{i}+9 \vec{j}, \quad \vec{y}=4 \vec{i}-7 \vec{j}, \quad \vec{z}=\vec{i}-3 \vec{j}-\vec{k} \end{aligned}$$ $$2 \vec{c}+\vec{x}$$

Short Answer

Expert verified
\( 21 \vec{j} \)

Step by step solution

01

Multiply Vector \( \vec{c} \) by 2

Given \( \vec{c} = \vec{i} + 6\vec{j} \). To find \( 2\vec{c} \), multiply each component by 2: \[ 2\vec{c} = 2(\vec{i} + 6\vec{j}) = 2\vec{i} + 12\vec{j}. \]
02

Add Vectors \( 2 \vec{c} \) and \( \vec{x} \)

We have \( 2\vec{c} = 2\vec{i} + 12\vec{j} \) and \( \vec{x} = -2\vec{i} + 9\vec{j} \). Add the corresponding components: \[ (2\vec{i} + 12\vec{j}) + (-2\vec{i} + 9\vec{j}) = (2 + (-2))\vec{i} + (12 + 9)\vec{j}. \]
03

Simplify the Expression

Simplify the expression: \( (2 - 2)\vec{i} + (12 + 9)\vec{j} = 0\vec{i} + 21\vec{j} = 21\vec{j} \). This is the resultant vector.

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Key Concepts

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

Vector Addition
In the realm of vector operations, vector addition is a fundamental process. It involves combining two or more vectors to result in a new vector.
The procedure for vector addition is straightforward:
  • Align the vectors so they share a common starting point.
  • Draw a new vector from the starting point to the tip of the final vector.
In our given example, vectors are expressed in terms of unit vectors \( \vec{i}, \vec{j}, \text{and } \vec{k} \). To add vectors like \( 2\vec{c} = 2\vec{i} + 12\vec{j} \) and \( \vec{x} = -2\vec{i} + 9\vec{j} \), you simply add corresponding components together:- Sum of \( \vec{i} \) components: \( 2 + (-2) = 0 \)- Sum of \( \vec{j} \) components: \( 12 + 9 = 21 \) This results in the vector \( 0\vec{i} + 21\vec{j} \), or just \( 21\vec{j} \).
Scalar Multiplication
Scalar multiplication involves multiplying a vector by a scalar, which is a real number.
This operation scales the vector, altering its magnitude but not its direction. For example, multiplying vector \( \vec{c} = \vec{i} + 6\vec{j} \) by the scalar 2 results in:
  • Each component of the vector is multiplied by the scalar.
  • \( \vec{i} \) becomes \( 2\vec{i} \).
  • \( 6\vec{j} \) becomes \( 12\vec{j} \).
Thus, \( 2\vec{c} = 2(\vec{i} + 6\vec{j}) = 2\vec{i} + 12\vec{j} \). Notice how the vector length doubled, yet the direction remained consistent with the original \( \vec{c} \) vector.
Resultant Vector
The resultant vector is the outcome of combining vectors through addition. In our example, after undertaking both scalar multiplication and vector addition, we achieved the vector \( 21\vec{j} \).
This resultant vector represents the total effect of the vectors involved in the operation. To determine the resultant vector:
  • Perform any necessary scalar multiplications first to modify vector magnitudes.
  • Proceed to address the vector additions by adding corresponding vector components.
  • The simplification yields the final vector, showing changes in direction or length.
Understanding the resultant vector is crucial. It indicates the comprehensive influence of multiple vectors in a given scenario.

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Most popular questions from this chapter

Perform the following operations on the given 3 -dimensional vectors.$$\vec{a}=2 \vec{j}+\vec{k} \quad \vec{b}=-3 \vec{i}+5 \vec{j}+4 \vec{k} \quad \vec{c}=\vec{i}+6 \vec{j}$$ $$\vec{y}=4 \vec{i}-7 \vec{j} \quad \vec{z}=\vec{i}-3 \vec{j}-\vec{k}$$ $$\vec{a} \cdot \vec{y}$$

Give reasons for your answer. The plane \(x+2 y-3 z=5\) has normal vector \(\vec{i}+2 \vec{j}-3 \vec{k}\)

List any vectors that are parallel to each other and any vectors that are perpendicular to each other: \(\vec{v}_{1}=\vec{i}-2 \vec{j} \quad \vec{v}_{2}=2 \vec{i}+4 \vec{j}\) \(\vec{v}_{3}=3 \vec{i}+1.5 \vec{j} \quad \vec{v}_{4}=-1.2 \vec{i}+2.4 \vec{j}\) \(\vec{v}_{5}=-5 \vec{i}-2.5 \vec{j} \quad \vec{v}_{6}=12 \vec{i}-12 \vec{j}\) \(\vec{v}_{7}=4 \vec{i}+2 \vec{j} \quad \vec{v}_{8}=3 \vec{i}-6 \vec{j}\) \(\vec{v}_{9}=0.70 \vec{i}-0.35 \vec{j}\)

A canoe is moving with velocity \(\vec{v}=5 \vec{i}+3 \vec{j} \mathrm{m} / \mathrm{sec}\) relative to the water. The velocity of the current in the water is \(\vec{c}=\vec{i}+2 \vec{j}\) m/sec. (a) What is the speed of the current? (b) What is the speed of the current in the direction of the canoe's motion?

Match the planes in (a)-(d) with one or more of the descriptions in (I)-(IV). No reasons are needed. (a) \(\quad 3 x-y+z=0\) (b) \(4 x+y+2 z-5=0\) (c) \(x+y=5\) (d) \(\quad x=5\) I Goes through the origin. II Has a normal vector parallel to the \(x y\) -plane. III Goes through the point (0,5,0) IV Has a normal vector whose dot products with \(\vec{i}, \vec{j}\) \(\vec{k}\) are all positive.
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