Answer

Verified

403.8k+ views

**Hint:**Start by considering the points as some variable and find out the direction ratios of the line joining these two points. Convert this direction ratios into either vector or cartesian format and apply relevant formula for line passing through the point in direction of AB , Convert one form to the other by taking the correct direction ratios and coordinates.

**Complete step-by-step answer:**

Given,

(3,-2,-5)= A(say)

(3,-2,6)= B(say)

Let the line passing through the points A and B be AB.

Now , we will find out the direction ratios of AB which can be found out by taking the difference of one coordinate to the other.

So, The direction ratios of AB will be

X coordinates=$l$ = $3 - 3 = 0$

Y coordinates =$m$ =$ - 2 - ( - 2) = - 2 + 2 = 0$

Z coordinates = $n$ =$6 - ( - 5) = 6 + 5 = 11$

Now , that we know direction ratios of AB ,let us write it in vector form

$\overrightarrow c = 0\hat i + 0\hat j + 11\hat k$

Since, AB passes through A(3,-2,-5), the position vector of A will be written as

$\overrightarrow a = 3\hat i - 2\hat j - 5\hat k$

The equation of AB in vector form is given by the relation

$\overrightarrow r $ = $\overrightarrow a $ + $ \lambda \overrightarrow c $

Substituting the values of $\overrightarrow a $and $\overrightarrow c $, we get

$\Rightarrow \overrightarrow r = (3\hat i - 2\hat j - 5\hat k) + \lambda 11\hat k$

The equation of AB in Cartesian form is given by the relation

$\dfrac{{x - {x_1}}}{l} = \dfrac{{y - {y_1}}}{m} = \dfrac{{z - {z_1}}}{n}$ where ${x_1},{y_1},{z_1}$are the coordinates of point passing through and $l,m,n$are the direction ratios of the line.

Substituting the values of coordinates of A and direction ratio of AB , we get

$ \Rightarrow \dfrac{{x - 3}}{0} = \dfrac{{y - \left( { - 2} \right)}}{0} = \dfrac{{z - \left( { - 5} \right)}}{{11}}$

$\dfrac{{x - 3}}{0} = \dfrac{{y + 2}}{0} = \dfrac{{z + 5}}{{11}}$

So, this is the required Cartesian equation.

Therefore , the equation of line AB in vector form is $\overrightarrow {AB} = (3\hat i - 2\hat j - 5\hat k) + \lambda 11\hat k$ and in the cartesian form is $\dfrac{{x - 3}}{0} = \dfrac{{y + 2}}{0} = \dfrac{{z + 5}}{{11}}$.

**Note:**Such similar questions can be solved using the above procedure. If the equation is found in one form whether vector or cartesian it can easily be converted into the other easily , by taking correct values. Attention must be given while substituting the values as it may lead to wrong answers.

Recently Updated Pages

What number is 20 of 400 class 8 maths CBSE

Which one of the following numbers is completely divisible class 8 maths CBSE

What number is 78 of 50 A 32 B 35 C 36 D 39 E 41 class 8 maths CBSE

How many integers are there between 10 and 2 and how class 8 maths CBSE

The 3 is what percent of 12 class 8 maths CBSE

Find the circumference of the circle having radius class 8 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

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

List some examples of Rabi and Kharif crops class 8 biology CBSE

Which are the Top 10 Largest Countries of the World?

The provincial president of the constituent assembly class 11 social science CBSE

Write the 6 fundamental rights of India and explain in detail