/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 38 If \((x, y, z) \neq(Q, 0,0)\) an... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 1: Problem 38

If \((x, y, z) \neq(Q, 0,0)\) and \((\hat{i}+\hat{j}+3 \hat{k}) x+(3 \hat{i}-3 \hat{j}+\hat{k}) y\) \(+(-4 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}) z=\lambda(x \hat{\mathbf{i}}+\hat{\mathbf{j}}+z \hat{\mathbf{k}})\), then the value of \(\lambda\) will be (a) \(-20\) (b) \(0,-2\) (c) \(-1,0\) (d) \(0-1\)

Short Answer

Expert verified
The value of \(\lambda\) is 0.

Step by step solution

01

Expand the Dot Product

Calculate the dot product on the left side: \[(\hat{i} + \hat{j} + 3 \hat{k}) x + (3 \hat{i} - 3\hat{j} + \hat{k}) y + ((-4) \hat{i} + 5 \hat{j}) z\]This simplifies to \[(x + 0y - 4z)\hat{i} + (y + 0x + 5z)\hat{j} + (3x + y)\hat{k}.\]
02

Expand the Right Side

The right side of the equation is given by:\[\lambda (x \hat{i} + \hat{j} + z \hat{k})\]This simplifies to: \[\lambda x \hat{i} + \lambda \hat{j} + \lambda z \hat{k}.\]
03

Equate the Coefficients

Now equate the coefficients of \(\hat{i}, \hat{j}, \text{and} \hat{k}\) from the left and right sides:1. For \(\hat{i}: (x - 4z) = \lambda x\)2. For \(\hat{j}: (y + 5z) = \lambda\)3. For \(\hat{k}: (3x + y) = \lambda z\).
04

Analyze the Coefficient Equations

Let's solve these equations for consistency and to find \(\lambda\):1. From \(x - 4z = \lambda x\): Rearrange to get \(x(1 - \lambda) = 4z\).2. From \(y + 5z = \lambda\).3. From \(3x + y = \lambda z\).For these equations to hold simultaneously for non-zero \((x, y, z)\), check for specific \(\lambda\) values.
05

Solve for \(\lambda\)

From the equation \(x(1 - \lambda) = 4z\), if \(x eq 0\), then \(1 - \lambda eq 0\) and \(z eq 0\). Rearrange gives \(z = \frac{x(1-\lambda)}{4}\).Substitute this in the \(\hat{k}\) equation:\[3x + y = \lambda \frac{x(1-\lambda)}{4}\]and this in \(\hat{j}\) equation, \(y + 5\frac{x(1-\lambda)}{4} = \lambda\).Try \(\lambda = 0\) and \(\lambda = 1\) to check for consistency. Only \(\lambda = 0\) satisfies all conditions without contradiction.

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Dot Product
The dot product, also known as the scalar product, is a fundamental operation in vector algebra. It relates two vectors and gives a scalar output.
  • The dot product of two vectors, say \( \mathbf{A} \) and \( \mathbf{B} \), is calculated by multiplying their corresponding components and summing up those products: \( \mathbf{A} \cdot \mathbf{B} = A_x B_x + A_y B_y + A_z B_z \).
  • Geometrically, it can be construed as the product of the magnitudes of the two vectors and the cosine of the angle between them.
  • In contexts like the given problem, the dot product helps simplify expressions involving vectors to make comparison straightforward.
Using the dot product simplifies the problem by breaking down complex vector expressions into recognizable patterns, making it easier to equate and solve for unknowns like \( \lambda \) in this case.
Vector Equation
A vector equation represents one vector as a combination of other vectors, suggesting a relationship among them. Being concisely written in vector form, these equations provide insights into the directions and magnitudes involved.
  • In the given problem, the left side is a sum of three vectors scaled by \( x, y, \) and \( z \), and is equated to a vector scaled by \( \lambda \).
  • Breaking down this vector equation requires matching coefficients of \( \hat{i}, \hat{j}, \) and \( \hat{k} \), which transforms complex vector expressions into simpler scalar equations.
  • This method allows determining how different vector components relate to each other, clarifying the roles of variables and constants.
Utilizing vector equations aids in visualizing the problem's structure, facilitating the transition from a complex vector representation to a manageable set of scalar equations.
System of Equations
A system of equations refers to a set of two or more equations with multiple variables that are solved together. In the context of our problem, each coefficient comparison leads to a separate equation forming a system:
  • For \( \hat{i} \): \( x - 4z = \lambda x \)
  • For \( \hat{j} \): \( y + 5z = \lambda \)
  • For \( \hat{k} \): \( 3x + y = \lambda z \)
Such systems require simultaneous solutions to find the values of variables such as \( \lambda \). Solving these involves substitution or elimination methods to systematically reduce and solve the equations. Supporting the process of isolating \( \lambda \) for consistency is a crucial step in understanding how all parts of the equation relate to each other for correct solutions.
Consistency of Equations
The consistency of a system of equations refers to whether there is at least one set of values for the variables that satisfies all the equations simultaneously. In our context:
  • Key to checking consistency is confirming that none of the equations contradict each other for the selected values of \( \lambda \).
  • If inconsistent, the system has no solution. Here, specific \( \lambda \) values were tested, revealing that only \( \lambda = 0 \) allows the system to remain consistent.
Consistency implies that every part of the vector equation correctly matches with independent vectorial and scalar parts when \( \lambda \) maintains integrity across the entire system. Confirming this ensures the derived and evaluated solutions are indeed viable under given conditions.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

If \(A, B\) and \(C\) are the vertices of a triangle whose position vectors are \(\mathbf{a}, \mathbf{b}\) and \(\mathbf{c}\) and \(G\) is the centroid of the \(\triangle A B C\), then \(\mathbf{G A}+\mathbf{G B}+\mathbf{G C}\) is (a) 0 (b) \(\mathrm{A}+\mathbf{B}+\mathrm{C}\) (c) \(\frac{a+b+c}{3}\) (d) \(\frac{a+b-c}{3}\)

If \(A B C D E F\) is regular hexagon, then \(\mathbf{A D}+\mathbf{E B}+\mathbf{F C}\) is equal to (a) 0 (b) \(2 \mathrm{AB}\) (c) \(3 \mathrm{AB}\) (d) \(4 \mathrm{AB}\)

If \(A B C D E F\) is a regular hexagon and \(\mathbf{A B}+\mathbf{A C}+\mathbf{A D}+\mathbf{A E}+\mathbf{A F}=\lambda \mathbf{A D}\), then \(\lambda\) is equal to (a) \(\underline{2}\) (b) 3 (c) 4 (d) 6

If a and bare position vector of two points \(A, B\) and \(C\) divides \(A B\) in ratio \(2: 1\), then position vector of \(C\) is (a) \(\frac{a+2 b}{3}\) (b) \(\frac{2 \mathrm{i}+\mathrm{b}}{3}\) (c) \(\frac{a+2}{3}\) (d) \(\frac{a+b}{2}\)

If the vectors \(\mathbf{A B}=3 \hat{\mathbf{i}}+4 \hat{\mathbf{k}}\) and \(\mathbf{A C}=5 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}\) are the sides of a \(\triangle A B C\), then the length of the median through \(A\) is [JEE Main 2013, 2003] (a) \(\sqrt{18}\) (b) \(\sqrt{72}\) (c) \(\sqrt{33}\) (d) \(\sqrt{45}\)
See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Calculus

Read Explanation

Logic and Functions

Read Explanation

Pure Maths

Read Explanation

Decision Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.