RD Sharma Class 12 Exercise 22.3 Algebra of Vectors Solutions Maths - Download PDF Free Online

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# RD Sharma Class 12 Exercise 22.3 Algebra of Vectors Solutions Maths - Download PDF Free Online

Edited By Satyajeet Kumar | Updated on Jan 24, 2022 06:19 PM IST

RD Sharma solutions have exercise-wise solutions for every chapter. For example, RD Sharma Class 12th Exercise 22.3 has seven inquiries alongside examples to simplify the excursion for each understudy. Moreover, position Vectors of the place of convergence, coplanar are discussed in this specific exercise which would be helpful for students.

Algebra-based math of vectors is one of the main chapters in Class 12 Mathematics. Here we concentrate on the variable-based math of vector amounts. Scalars and vectors are two kinds of actual quantities. RD Sharma solutions have come to help students who face trouble in this chapter.

## Algebra of Vectors Excercise: 22.3

Algebra of Vectors Exercise 22.3 Question 1

$\frac{5}{3}\vec{a},3\vec{a}+4\vec{b}$
Hint: if a point R divides a line PQ into m:n internally then point $P=\frac{m\vec{OR+n\vec{OP}}}{m-n}$and externally then point $R=\frac{m\vec{OQ+n\vec{OP}}}{m-n}$
Given:
$\vec{OP}=2\vec{a}+\vec{b}$
$\vec{OQ}=\vec{a}-2\vec{b}$
Solution:

$(i) \overrightarrow{O R}( internally )$
$=\frac{1(\vec{a}-2 \vec{b})+2(2 \vec{a}+\vec{b})}{1+2}$
$=\frac{\vec{a}-2 \vec{b}+4 \vec{a}+2 \vec{b}}{3}$
$=\frac{5 \vec{a}}{3}$
$(ii) \overrightarrow{O R}( Externally )$
$=\frac{1(\vec{a}-2 \vec{b})-2(2 \vec{a}+\vec{b})}{1-2}$
$=\frac{\vec{a}-2 \vec{b}-4 \vec{b}-2 \vec{b}}{-1}$
$=\frac{-3 \vec{a}-4 \vec{b}}{-1}$
$=3 \vec{a}+4 \vec{b}$

Algebra of Vectors Exercise 22.3 Question 2

$-\vec{a}-7\vec{b}$
Hint: $if \: a \: point \: Z \: divides\: a \: line\: XY \: into\: m:n\: ratio \: externally \: then \: point$$Z=\frac{m\vec{OY-n\vec{OX}}}{m-n}$
Given:
$\vec{OX}=3\vec{a}+\vec{b}$
$\vec{OY}=\vec{a}-3\vec{b}$
Solution: $\vec{d}$

$\vec{OZ}\: (Externally)$
\begin{aligned} &=\frac{(2(\vec{a}-3 \vec{b})-1(3 \vec{a}+\vec{b}))}{2-1} \\ &=(2 \vec{a}-6 \vec{b}-3 \vec{a}-\vec{b}) \\ &=-\vec{a}-7 \vec{b} \end{aligned}

Algebra of Vectors Exercise 22.3 Question 4

Answer: $3\vec{b}-2\vec{a},2\vec{a}-\vec{b}$
Hint: Vector law of addition, (two vectors one represented as two sides of the triangle with the order of magnitude and direction, then the third side of the triangle represents the magnitude and direction of the resultant vector.)
Given: $\vec{AC}=3\vec{AB}$
Solution:

$In \triangle \mathrm{OAB},$
$=\overrightarrow{O A}+\overrightarrow{A B}=\overrightarrow{O B}\: (By \: triangle\: law \: of \: vector \: addition)$
$\mathrm{AB}=\vec{b}-\vec{a}$
$In \triangle \mathrm{OAC}$
$=\overrightarrow{O C}=\overrightarrow{O A}+\overrightarrow{A C}$ $(By\: triangle \: law \: of\: vector \: addition)$

$=\overrightarrow{O A}+3 \overrightarrow{A B} \$ $[ given \overrightarrow{A C}=3 \overrightarrow{A B}]$
$=\vec{a}+3(\vec{b}-\vec{a})$
$=3 \vec{b}-2 \vec{a}$
$\text{Again in }\triangle O A D$
$\overrightarrow{O D}=\overrightarrow{O A}+\overrightarrow{A D} \ \ \ (\text{ By triangle law of vector addition})$
$=\vec{a}+\vec{a}-\vec{b} \quad[\because \overrightarrow{A D}=-\overrightarrow{A B}]$
$=2 \vec{a}-\vec{b}$

Algebra of Vectors Exercise 22.3 Question 5

Answer:\begin{aligned} &\frac{3 \vec{a}+5 \vec{c}}{8}=\frac{2 \vec{b}+6 \vec{d}}{8} \\ \end{aligned}
Hint: Use vector algebra
Given:\begin{aligned} &\qquad 3 \vec{a}-2 \vec{b}+5 \vec{c}-6 \vec{d}=0 \\ \end{aligned}
Solution: $\qquad 3\vec{a} +5 \vec{c}=2 \vec{b}+6\vec{d}$
\begin{aligned} &\frac{3 \vec{a}+5 \vec{c}}{8}=\frac{2 \vec{b}+6 \vec{d}}{8} \end{aligned}

AC and BD are intersect at the point.

A,B,C,D are co- planner.
P.V of point of intersection of line segment AC and BD are \begin{aligned} &\frac{3 \vec{a}+5 \vec{c}}{8}\: or\: \frac{2 \vec{b}+6 \vec{d}}{8} \\ \end{aligned}

Algebra of Vectors Exercise 22.3 Question 6

Answer:$\frac{-5 p+6 \vec{r}}{11} o r \frac{2 \vec{q}+9 \vec{s}}{11}$

Hint: Use vector algebra.
Given: $5 \vec{p}-2 \vec{q}+6 \vec{r}-9 \vec{s}=0$
Solution: $5 \vec{p}-6 \vec{r}=2 \vec{q}-9 \vec{s}$
$\frac{-5 p+6 \vec{r}}{11} =\frac{2 \vec{q}+9 \vec{s}}{11}$....(1)

As we know that, if the point C divides the line segment AB in the ratio m:n the $\overrightarrow{O C}=\frac{n \overrightarrow{O A}+m \overrightarrow{O B}}{m+n}$
let AB the point which PR in the ratio 6:5 and B be the point which divides QS in the ratio 9:2 .hence the P.V of AB are $\frac{6 \vec{p}+6 \vec{r}}{11}, \frac{2 \vec{q}+9 \vec{s}}{11}$
From (1)
The P.V of AB is same.
= point A and B intersect at the same point
= PQ and QS intersect at A
of two line segment intersect then it must be Co-planner.
P,Q,R,S are co-planner.

Algebra of Vectors Exercise 22.3 Question 7

Answer: – we need to prove that $I=\frac{\alpha \vec{a}+(\beta+\gamma) \vec{d}}{\alpha+\beta+\gamma}$
Hint: Use vector algebra.
Given: The vertices A,B,C of triangle ABC have respectively. P.V $\vec{a},\vec{b},\vec{c}$ with respect to the given origin 0.
Solution :

Let the PV of A,B and C w.r.t some origin 0 be $\vec{a},\vec{b},\vec{c}$ respectively. let DB be a point on BC where bisector of angle D meets.
let $\vec{d}$ be P.V of D which divide CB internally in the ratio β and $\gamma$ where $\beta =\mid \vec{AC}\mid$ and $\gamma =\mid \vec{AB}\mid$
Thus $\beta =\mid \vec{c}-\vec{a}\mid$ and $\gamma =\mid \vec{b}-\vec{a}\mid$by section formula, the P.V of D is given by
\begin{aligned} &=\overrightarrow{O D}=\frac{\beta \vec{b}+\gamma \vec{c}}{\beta+\gamma} \\ &=\alpha=|\vec{b}-\vec{c}| \end{aligned}

In centre is the concurrent point angle bisector and in centre divides the line AD in the ratio
$=a=\beta -\gamma$
So the P.V of in centre is given by $I=\frac{\alpha \vec{a}+(\beta+\gamma) \vec{d}}{\alpha+\beta+\gamma}$

A team of experts prepares Class 12th RD Sharma Chapter 22 Exercise 22.3 Solutions using the most recent NCERT textbook and marking schemes. When a student refers to RD Sharma Class 12 Solutions Algebra of Vectors Ex. 22.3, all concepts are clear, and the student understands the exams better. When a student refers to RD Sharma Class 12 Solutions Chapter 22 ex 22.3, the following benefits will accrue:

1. Expertly crafted:

Subject experts create RD Sharma Class 12th Chapter 22 Exercise 22.3 in accordance with the most recent marking pattern and CBSE guidelines. Every student must practice RD Sharma Class 12th Exercise 22.3 to achieve a high score. These solutions explain the topics mentioned above so that even a newcomer can learn and solve them.

2. Efficient

RD Sharma Class 12th Exercise 22.3 offers the best in class solutions for students. It is updated to the latest version of the book and contains all the new concepts. As it follows the CBSE syllabus, students can use it to prepare efficiently and save time. Moreover, referring to this material will help them gain more knowledge on the subject and understand different aspects of the chapter.

3. Easy to understand

The solutions provided in this material contain step-by-step answers which students can refer to increase their knowledge. Students can learn about various alternatives to solving a problem and choose the one which best suits them.

4. Exam-oriented

RD Sharma Class 12 Solutions Ex. 22.3 material contains solutions that follow the CBSE syllabus and are exam-oriented. As RD Sharma books are widely used in schools, this material's questions can also appear in exams. This is an efficient and paperless alternative for preparation.

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## RD Sharma Chapter wise Solutions

1. What are the Different Kinds of Vector Multiplication?

Vector multiplication is classified into two types:

Dot Product: P • Q = |P| |Q| cos θ

Cross Product: P × Q = |P| |Q| sin θ

2. What is the vector's magnitude?

Vertical lines on both sides of the given vector "|a|" show the magnitude of a vector. It represents the vector's length.

3. What are some vector examples?

Force, velocity, acceleration, and other quantities with both magnitude and direction are examples of vectors.

4. Are there any hidden charges for this material?

No, this material is absolutely free on courier 360's website with no added costs.

5. Can I finish my homework through this material?

As this material follows the CBSE syllabus, students can use it to practice and finish their homework.

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##### Fashion Journalist

Fashion journalism involves performing research and writing about the most recent fashion trends. Journalists obtain this knowledge by collaborating with stylists, conducting interviews with fashion designers, and attending fashion shows, photoshoots, and conferences. A fashion Journalist  job is to write copy for trade and advertisement journals, fashion magazines, newspapers, and online fashion forums about style and fashion.

2 Jobs Available
##### Multimedia Specialist

A multimedia specialist is a media professional who creates, audio, videos, graphic image files, computer animations for multimedia applications. He or she is responsible for planning, producing, and maintaining websites and applications.

2 Jobs Available
##### Corporate Executive

Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

2 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### Quality Controller

A quality controller plays a crucial role in an organisation. He or she is responsible for performing quality checks on manufactured products. He or she identifies the defects in a product and rejects the product.

A quality controller records detailed information about products with defects and sends it to the supervisor or plant manager to take necessary actions to improve the production process.

3 Jobs Available
##### Production Manager

Production Manager Job Description: A Production Manager is responsible for ensuring smooth running of manufacturing processes in an efficient manner. He or she plans and organises production schedules. The role of Production Manager involves estimation, negotiation on budget and timescales with the clients and managers.

3 Jobs Available
##### QA Manager

Quality Assurance Manager Job Description: A QA Manager is an administrative professional responsible for overseeing the activity of the QA department and staff. It involves developing, implementing and maintaining a system that is qualified and reliable for testing to meet specifications of products of organisations as well as development processes.

2 Jobs Available

A QA Lead is in charge of the QA Team. The role of QA Lead comes with the responsibility of assessing services and products in order to determine that he or she meets the quality standards. He or she develops, implements and manages test plans.

2 Jobs Available
##### Reliability Engineer

Are you searching for a Reliability Engineer job description? A Reliability Engineer is responsible for ensuring long lasting and high quality products. He or she ensures that materials, manufacturing equipment, components and processes are error free. A Reliability Engineer role comes with the responsibility of minimising risks and effectiveness of processes and equipment.

2 Jobs Available
##### Safety Manager

A Safety Manager is a professional responsible for employee’s safety at work. He or she plans, implements and oversees the company’s employee safety. A Safety Manager ensures compliance and adherence to Occupational Health and Safety (OHS) guidelines.

2 Jobs Available
##### Corporate Executive

Are you searching for a Corporate Executive job description? A Corporate Executive role comes with administrative duties. He or she provides support to the leadership of the organisation. A Corporate Executive fulfils the business purpose and ensures its financial stability. In this article, we are going to discuss how to become corporate executive.

2 Jobs Available
##### ITSM Manager

ITSM Manager is a professional responsible for heading the ITSM (Information Technology Service Management) or (Information Technology Infrastructure Library) processes. He or she ensures that operation management provides appropriate resource levels for problem resolutions. The ITSM Manager oversees the level of prioritisation for the problems, critical incidents, planned as well as proactive tasks.

3 Jobs Available
##### Information Security Manager

Individuals in the information security manager career path involves in overseeing and controlling all aspects of computer security. The IT security manager job description includes planning and carrying out security measures to protect the business data and information from corruption, theft, unauthorised access, and deliberate attack

3 Jobs Available
##### Computer Programmer

Careers in computer programming primarily refer to the systematic act of writing code and moreover include wider computer science areas. The word 'programmer' or 'coder' has entered into practice with the growing number of newly self-taught tech enthusiasts. Computer programming careers involve the use of designs created by software developers and engineers and transforming them into commands that can be implemented by computers. These commands result in regular usage of social media sites, word-processing applications and browsers.

3 Jobs Available
##### Product Manager

A Product Manager is a professional responsible for product planning and marketing. He or she manages the product throughout the Product Life Cycle, gathering and prioritising the product. A product manager job description includes defining the product vision and working closely with team members of other departments to deliver winning products.

3 Jobs Available
##### IT Consultant

An IT Consultant is a professional who is also known as a technology consultant. He or she is required to provide consultation to industrial and commercial clients to resolve business and IT problems and acquire optimum growth. An IT consultant can find work by signing up with an IT consultancy firm, or he or she can work on their own as independent contractors and select the projects he or she wants to work on.

2 Jobs Available
##### Data Architect

A Data Architect role involves formulating the organisational data strategy. It involves data quality, flow of data within the organisation and security of data. The vision of Data Architect provides support to convert business requirements into technical requirements.

2 Jobs Available
##### Security Engineer

The Security Engineer is responsible for overseeing and controlling the various aspects of a company's computer security. Individuals in the security engineer jobs include developing and implementing policies and procedures to protect the data and information of the organisation. In this article, we will discuss the security engineer job description, security engineer skills, security engineer course, and security engineer career path.

2 Jobs Available
##### UX Architect

A UX Architect is someone who influences the design processes and its outcomes. He or she possesses a solid understanding of user research, information architecture, interaction design and content strategy.

2 Jobs Available