The lines whose vector equations are r = 2 hati - 3 hatj + 7 hatk + λ (2 hati + p hatj + 5 hatk) and r = hati - 2 hatj + 3 hatk + μ (3 hati + p hatj + p hatk) are perpendicular for all values of λ and μ if p =

Question

The lines whose vector equations are r = 2 hati - 3 hatj + 7 hatk + λ (2 hati + p hatj + 5 hatk) and r = hati - 2 hatj + 3 hatk + μ (3 hati + p hatj + p hatk) are perpendicular for all values of λ and μ if p = (A) 1 (B) -1 (C) -6 (D) 6

Tardigrade Admin · Accepted Answer

2 hati - p hatj+ 5 hatk and 3 hati + p hatj + p hatk are perpendicular ⇒ 2 × 3 + p (-p) + 5 (p) = 0 ⇒ p2 - 5p - 6 = 0 ⇒ p = -1 or p = 6 Hence for p = 6, the lines are perpendicular.

Q. The lines whose vector equations are
$r = 2 \hat{i} - 3 \hat{j} + 7 \hat{k} + λ (2 \hat{i} + p \hat{j} + 5 \hat{k})$
and $r = \hat{i} - 2 \hat{j} + 3 \hat{k} + μ (3 \hat{i} + p \hat{j} + p \hat{k})$
are perpendicular for all values of $λ$ and $μ$ if p =

1

-1

-6

6

$2 \hat{i} - p \hat{j} + 5 \hat{k}$ and $3 \hat{i} + p \hat{j} + p \hat{k}$ are perpendicular
$\Rightarrow 2 \times 3 + p (- p) + 5 (p) = 0$
$\Rightarrow p^{2} - 5 p - 6 = 0 \Rightarrow p = - 1$ or $p = 6$
Hence for p = 6, the lines are perpendicular.

Q. The lines whose vector equations are r=2i^−3j^​+7k^+λ(2i^+pj^​+5k^) and r=i^−2j^​+3k^+μ(3i^+pj^​+pk^) are perpendicular for all values of λ and μ if p =

1

-1

-6

6

2i^−pj^​+5k^ and 3i^+pj^​+pk^ are perpendicular ⇒2×3+p(−p)+5(p)=0 ⇒p2−5p−6=0⇒p=−1 or p=6 Hence for p = 6, the lines are perpendicular.

Q. The lines whose vector equations are
$r = 2 \hat{i} - 3 \hat{j} + 7 \hat{k} + λ (2 \hat{i} + p \hat{j} + 5 \hat{k})$
and $r = \hat{i} - 2 \hat{j} + 3 \hat{k} + μ (3 \hat{i} + p \hat{j} + p \hat{k})$
are perpendicular for all values of $λ$ and $μ$ if p =

$2 \hat{i} - p \hat{j} + 5 \hat{k}$ and $3 \hat{i} + p \hat{j} + p \hat{k}$ are perpendicular
$\Rightarrow 2 \times 3 + p (- p) + 5 (p) = 0$
$\Rightarrow p^{2} - 5 p - 6 = 0 \Rightarrow p = - 1$ or $p = 6$
Hence for p = 6, the lines are perpendicular.