If the line through the points A(k, 1,-1) and B (2 k, 0,2) is perpendicular to the line through the points B and C(2+2 k, k, 1), then what is the value of k ?

Question

If the line through the points A(k, 1,-1) and B (2 k, 0,2) is perpendicular to the line through the points B and C(2+2 k, k, 1), then what is the value of k ? (A) -1 (B) 1 (C) -3 (D) 3

Tardigrade Admin · Accepted Answer

A(k, 1,-1) B(2 k, 0,2),C(2+2 k, k, 1) A B d r' S=(2 k-k, 0-1,2-(-1) =(k,-1,3) B C d r' S=(2+2 k-2 k, k-0,1-2) =(2, k,-1) So a1 a2+b1 b2+c1 c2=0 k(2)+k(-1)+3(-1)=0 2 k+k-3=0 k=3

Q. If the line through the points $A (k, 1, - 1)$ and $B$ $(2 k, 0, 2)$ is perpendicular to the line through the points $B$ and $C (2 + 2 k, k, 1)$ , then what is the value of $k$ ?

-1

1

-3

3

$A (k, 1, - 1) B (2 k, 0, 2), C (2 + 2 k, k, 1)$
$A B$ d r' $S = (2 k - k, 0 - 1, 2 - (- 1)$
$= (k, - 1, 3)$
$BC$ d r' $S = (2 + 2 k - 2 k, k - 0, 1 - 2)$
$= (2, k, - 1)$
So $a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$
$k (2) + k (- 1) + 3 (- 1) = 0$
$2 k + k - 3 = 0$
$k = 3$

Q. If the line through the points A(k,1,−1) and B (2k,0,2) is perpendicular to the line through the points B and C(2+2k,k,1), then what is the value of k ?

-1

1

-3

3

A(k,1,−1)B(2k,0,2),C(2+2k,k,1) AB d r' S=(2k−k,0−1,2−(−1) =(k,−1,3) BC d r' S=(2+2k−2k,k−0,1−2) =(2,k,−1) So a1​a2​+b1​b2​+c1​c2​=0 k(2)+k(−1)+3(−1)=0 2k+k−3=0 k=3

Q. If the line through the points $A (k, 1, - 1)$ and $B$ $(2 k, 0, 2)$ is perpendicular to the line through the points $B$ and $C (2 + 2 k, k, 1)$ , then what is the value of $k$ ?

$A (k, 1, - 1) B (2 k, 0, 2), C (2 + 2 k, k, 1)$
$A B$ d r' $S = (2 k - k, 0 - 1, 2 - (- 1)$
$= (k, - 1, 3)$
$BC$ d r' $S = (2 + 2 k - 2 k, k - 0, 1 - 2)$
$= (2, k, - 1)$
So $a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$
$k (2) + k (- 1) + 3 (- 1) = 0$
$2 k + k - 3 = 0$
$k = 3$