Show that the vectors 2-k- -3-5k- and 3-4-4k- form the vertices of a right angled triangle -

Let vectors #160; 2î ĵ+k̂, #160;î 3ĵ 5#10;k and 3î 4ĵ 4k̂ be position vectors of #10;points A, B and C respectively.i.e., O⃗A ...

You visited us 1 times! Enjoying our articles? Unlock Full Access!

Byju's Answer

Standard XII

Mathematics

Triangle in Rectangular Cartesian Coordinates

Show that the...

Question

Show that the vectors $2^i -^j +^k,^i - 3^j - 5^k$ and $3^i - 4^j - 4^k$ form the vertices of a right angled triangle .

Open in App

Solution

Let vectors $2^i -^j +^k,^i - 3^j - 5^k$ and $3^i - 4^j - 4^k$ be position vectors of points $A, B$ and $C$ respectively.
i.e., $\to O A = 2^i -^j +^k, \to O B =^i - 3^j - 5^k$ and $\to O C = 3^i - 4^j - 4^k$
Now, vectors $\to A B, \to B C$ , and $\to A C$ represent the sides of $Δ A B C$ .
$∴ \to A B = (1 - 2)^i + (- 3 + 1)^j + (- 5 - 1)^k = -^i - 2^j - 6^k$
$\to B C = (3 - 1)^i + (- 4 + 3)^j + (- 4 + 5)^k = 2^i -^j +^k$
$\to A C = (2 - 3)^i + (- 1 + 4)^j + (1 + 4)^k = -^i + 3^j + 5^k$
$| \to A B | = \sqrt{(- 1)^{2} + (- 2)^{2} + (- 6)^{2}} = \sqrt{1 + 4 + 36} = \sqrt{41}$
$| \to B C | = \sqrt{2^{2} + (- 1)^{2} + 1^{2}} = \sqrt{4 + 1 + 1} = \sqrt{6}$
$| \to A C | = \sqrt{(- 1)^{2} + 3^{2} + 5^{2}} = \sqrt{1 + 9 + 25} = \sqrt{35}$
$∴ | \to B C |^{2} + | \to A C |^{2} = 6 + 35 = 41 = | \to A B |^{2}$
Hence, $Δ A B C$ is a right-angled triangle.

Suggest Corrections

Similar questions

Q. Show that vectors

2^i -^k,^i - 3^j - 5^k

and

3^i - 4^j - 4^k

form the vertices of the triangle.

Q. Show that the vectors

2^i -^j +^k,^i - 3^j - 5^k a n d 3^i - 4^j - 4^k

form the vertices of a right angled triangle.

Show that the points A,B and C with position vectors, $a = 3^i - 4^j - 4^k, b = 2^i -^j +^k a n d c =^i - 3^j - 5^k$ respectively, form the vertices of a right angled triangle.

Q. Show that the vectors

a = 3^i - 2^j +^k, b =^i - 3^j + 5^k

and

c = 2^i +^j - 4^k

from a right angled triangle.

Q. Show that the points

A, B, C

with position vectors

2^i -^j +^k,^i - 3^j - 5^k

and

3^i - 4^j - 4^k

respectively, are the vertices of a right-angled triangle. Hence, find the area of the triangle.

Join BYJU'S Learning Program

Show that the vectors 2^i−^j+^k,^i−3^j−5^k and 3^i−4^j−4^k form the vertices of a right angled triangle .

Show that the vectors $2^i -^j +^k,^i - 3^j - 5^k$ and $3^i - 4^j - 4^k$ form the vertices of a right angled triangle .