In a triangle A B C , right angle at vertex A , if the position vectors of A , B and C are respectively 3 i ^ +   j ^ -   k ^ ,   - i ^ + 3 j ^ + p k ^ and 5 i ^ + q j ^ - 4 k ^ , then the point p , q lies on a line:

Question

In a triangle A B C , right angle at vertex A , if the position vectors of A , B and C are respectively 3 i ^ + &#160; j ^ - &#160; k ^ , &#160; - i ^ + 3 j ^ + p k ^ and 5 i ^ + q j ^ - 4 k ^ , then the point p , q lies on a line:

Aicharya · Accepted Answer

A B &#8594; = &#160; - 4 i ^ + 2 j ^ + p + 1 k ^ A C &#8594; = &#160; 2 i ^ + q - 1 j ^ - 3 k ^ A B &#8594; &#8869; A C &#8594; &#8658; A B &#8594; &#160; &#183; &#160; A C &#8594; = 0 - 8 + 2 q - 1 - 3 p + 1 = 0 3 p - 2 q + 13 = 0 p , q lies on 3 x - 2 y + 13 = 0 Slope = 3 2 &#8756;&#160;&#160; The required line makes an acute angle with the positive direction of x - axis.

In a triangle A B C , right angle at vertex A , if the position vectors of A , B and C are respectively 3 i…

Solution