Define Collinear Vectors Class 12 Maths CBSE - Vedantu

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AnswerQuestion Answers for Class 12Class 12 BiologyClass 12 ChemistryClass 12 EnglishClass 12 MathsClass 12 PhysicsClass 12 Social ScienceClass 12 Business StudiesClass 12 EconomicsQuestion Answers for Class 11Class 11 EconomicsClass 11 Computer ScienceClass 11 BiologyClass 11 ChemistryClass 11 EnglishClass 11 MathsClass 11 PhysicsClass 11 Social ScienceClass 11 AccountancyClass 11 Business StudiesQuestion Answers for Class 10Class 10 ScienceClass 10 EnglishClass 10 MathsClass 10 Social ScienceClass 10 General KnowledgeQuestion Answers for Class 9Class 9 General KnowledgeClass 9 ScienceClass 9 EnglishClass 9 MathsClass 9 Social ScienceQuestion Answers for Class 8Class 8 ScienceClass 8 EnglishClass 8 MathsClass 8 Social ScienceQuestion Answers for Class 7Class 7 ScienceClass 7 EnglishClass 7 MathsClass 7 Social ScienceQuestion Answers for Class 6Class 6 ScienceClass 6 EnglishClass 6 MathsClass 6 Social ScienceQuestion Answers for Class 5Class 5 ScienceClass 5 EnglishClass 5 MathsClass 5 Social ScienceQuestion Answers for Class 4Class 4 ScienceClass 4 EnglishClass 4 MathsDefine collinear vectors.AnswerVerified560.4k+ viewsHint: We start solving by recalling the definition of collinear vectors that they line on the same line or parallel lines. We use the fact that the components of one of the collinear vectors is equal to the multiples of another vector. We use the fact that the cross product of a collinear vector is zero to prove all the conditions about the collinear vectors.Complete step-by-step answer:Collinear vectors: - Vectors parallel to one or lying on one line are called collinear vectors.Condition of collinearity: - Two vectors are collinear if any of these conditions are done.Condition-1:- Two vectors a, b are collinear if there exists a number such that the below equation will become true. $\bar{a}=n.\bar{b}$Condition-2:- Two vectors are collinear if the relation of their coordinates are equal.This is not valid if one of the components is zero.Condition-3:- Two vectors are collinear if their cross product is equal to the zero vector.This is valid only in the case where 2 vectors are three-dimensional (spatial) vectors.Cross-product:- Cross product of vector a by vector b is the vector c, the length of which is numerically equal to the area of parallelogram constructed on vector a, b, direction is perpendicular to the plane of the vectors of a, b. If a, b vectors are written as $xi+yj+zk;\text{ pi+qj+rk}$, we get cross product a, b represented by $a\times b$ as:$a\times b=\left|\begin{matrix} &i &j &k \\ &x &y &z \\ &p &q &r \\ \end{matrix} \right|$Apply this definition to condition-3 we get:Cross product a, b is 0. From condition 1, we get:$a=nb$. If $b=xi+yj+zk,$ we get value of a as,$a=nxi+nyj+nzk$.Cross product of $a\times b$ is written as:$a\times b=\left| \begin{matrix} &i &j &k \\ &nx &ny &nz \\ &x &y &z \\ \end{matrix} \right|$By expanding this, we get it as follows:\[\begin{align} & a\times b=\left( nzy-nzy \right)i-\left( nxz-nxz \right)j+\left( nxy-nxy \right)k \\ & a\times b=oi-oj+ok\text{ = zero vector}\text{.} \\ \end{align}\]Hence proved.Note: Be careful with the second condition. If a term is zero in one vector that condition will go wrong. While proving the condition-3 we must take care of a, b. Alternately, assume a as (x, y, z), by this you get b as $\left( \dfrac{x}{n},\dfrac{y}{n},\dfrac{z}{n} \right)$ . Substitute these, anyways you get the same answer.Recently Updated PagesMaster Class 12 English: Engaging Questions & Answers for SuccessMaster Class 12 Business Studies: Engaging Questions & Answers for SuccessMaster Class 12 Economics: Engaging Questions & Answers for SuccessMaster Class 12 Social Science: Engaging Questions & Answers for SuccessMaster Class 12 Maths: Engaging Questions & Answers for SuccessMaster Class 12 Chemistry: Engaging Questions & Answers for SuccessMaster Class 12 English: Engaging Questions & Answers for SuccessMaster Class 12 Business Studies: Engaging Questions & Answers for SuccessMaster Class 12 Economics: Engaging Questions & Answers for SuccessMaster Class 12 Social Science: Engaging Questions & Answers for SuccessMaster Class 12 Maths: Engaging Questions & Answers for SuccessMaster Class 12 Chemistry: Engaging Questions & Answers for Success

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