Question

Vectors 2i-j+k and i-3j-5k adn 3i-4j-4k , join the origin of a system of rectangular coor-dinates to points A,B and C respectively. Show that ABC is a right angled triangle.​

Answer 1

Answer:

Hey mate! Here is your proof.

Explanation:

So let us consider,

$A(2i - j + k), \\B( i - 3j -5k) ,\\ C( 3i - 4j - 4k)$

We are aware that two vectors are perpendicular to each other if their scaler product is 0.

$AB = ( 1 - 2 )i + ( -3 - 1 ) j + ( -5 - 1 ) k = -i - 2j - 6k$

$and |AB|$ = √${ (-1) 2 + (-2) 2 + (-6) 2 } =$ √$(1 + 4 + 36) =$√41

Simplifying even further, we get,

$BC = (3 - 1)i + (-4 + 3 )j + (-4 + 5 )k = 2i - j + k$

Now,

$|BC| =$ √${(2)2 + (-1)2 + 12 } =$√$(4 + 1 + 1) =$ √$6$

$and CA = (2 - 3)i + (-1 + 4)j + (1 + 4)k = -i + 3j + 5k$

Now we get that,

$|CA| =$√${(-1)2 + 32 + 52 } =$ √$(1 + 9 + 25) =$√$35$

Again,

$|BC|2 + |CA|2 = 6 + 35 = 41 = |AB|2$

From the above information, we can finally say that,

So, triangle ABC is a right angle triangle.

Hope it helps you!