Answer

Verified

393.3k+ views

**Hint:**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 Pages

The base of a right prism is a pentagon whose sides class 10 maths CBSE

A die is thrown Find the probability that the number class 10 maths CBSE

A mans age is six times the age of his son In six years class 10 maths CBSE

A started a business with Rs 21000 and is joined afterwards class 10 maths CBSE

Aasifbhai bought a refrigerator at Rs 10000 After some class 10 maths CBSE

Give a brief history of the mathematician Pythagoras class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Name 10 Living and Non living things class 9 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Write the 6 fundamental rights of India and explain in detail