# Show that the line through points (1, -1, 2), (3, 4, -2) is perpendicular to the line through the points (0, 3, 2) and (3, 5, 6).

Answer

Verified

325.8k+ views

**Hint:**First see the equation of line passing through the points or see the direction ratio of the lines of two lines through the points given above and then use the condition of two perpendicular lines. Use the concept of dot and cross product.

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

First let's name the points given, A= (1, -1, 2) and B = (3, 4, -2), C = (0, 3, 2) and D = (3, 5, 6)

Two lines with direction ratio \[{a_1},{b_1},{c_1}\] and \[{a_2},{b_2},{c_2}\] are said to be perpendicular to each other if

\[{a_1}{a_2} + {b_1}{b_2} + {c_1}{c_2} = 0\]………(1)

Direction ratio of the line passing through the points is given by \[({x_2} - {x_1}),({y_{_2}} - {y_1}),({z_2} - {z_1})\]

∴ Direction ratio of the line passing through the points A and B is

\[(3 - 1),(4 - ( - 1)),( - 2 - 2)\]

\[ \Rightarrow \]2, 5, -4

∴\[{a_1}\] = 2, \[{b_1}\] = 5, \[{c_1}\]=-4

Direction ratio of the line passing through the points C and D is

\[(3 - 0),(5 - 3),(6 - 2)\]

\[ \Rightarrow \]3, 2, 4

\[ \Rightarrow \]\[{a_2}\] = 3, \[{b_2}\] = 2, \[{c_2}\] = 4

Now using equation (1) we get

= (2×3) + (5×2) + (-4×-4)

=6 + 10 + (-16)

=16 – 16

= 0.

**Since the equation (1) is satisfied which shows that the two lines passing through the above points are perpendicular to each other.**

**Note:**This question can also be solved using the dot product, for which you first have to make these points into vector equations of two lines and then the dot product of the two lines which should be equal to 0 to show that they are perpendicular.

Last updated date: 28th May 2023

•

Total views: 325.8k

•

Views today: 3.83k

Recently Updated Pages

If ab and c are unit vectors then left ab2 right+bc2+ca2 class 12 maths JEE_Main

A rod AB of length 4 units moves horizontally when class 11 maths JEE_Main

Evaluate the value of intlimits0pi cos 3xdx A 0 B 1 class 12 maths JEE_Main

Which of the following is correct 1 nleft S cup T right class 10 maths JEE_Main

What is the area of the triangle with vertices Aleft class 11 maths JEE_Main

KCN reacts readily to give a cyanide with A Ethyl alcohol class 12 chemistry JEE_Main

Trending doubts

What was the capital of Kanishka A Mathura B Purushapura class 7 social studies CBSE

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Tropic of Cancer passes through how many states? Name them.

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

Name the Largest and the Smallest Cell in the Human Body ?