If the planes r → · 2 i ^ - λ j ^ + 3 k ^ = 0 and r → · λ i ^ + 5 j ^ - k ^ = 5 are perpendicular to each other, then the value of λ 2 + λ is

We have, r → · λ i ^ + 5 j ^ - k ^ = 5 r → · 2 i ^ - λ j ^ + 3 k ^ = 0 We know that two planes r → · n → 1 = d 1 and r → · n → 2 = d 2 are perpendicular if n → 1 · n → 2 = 0 Therefore, λ i ^ + 5 j ^ - k ^ · 2 i ^ - λ j ^ + 3 k ^ = 0 ⇒ 2 λ - 5 λ - 3 = 0 ⇒ - 3 λ - 3 = 0 ⇒ λ = - 1 Therefore, λ 2 + λ = 0

Mathematics - 3D-Coordinate Geometry Question with Solution | TestHub

Mathematics3D-Coordinate GeometryMiscellaneous/MixedEasy2 minPYQ_2018

MathematicsEasysingle choice

If the planes $\vec{r} \cdot (2 \hat{i} - λ \hat{j} + 3 \hat{k}) = 0$ and $\vec{r} \cdot (λ \hat{i} + 5 \hat{j} - \hat{k}) = 5$ are perpendicular to each other, then the value of $λ^{2} + λ$ is

Options:

Answer:

Solution:

We have,

$\vec{r} \cdot (λ \hat{i} + 5 \hat{j} - \hat{k}) = 5$

$\vec{r} \cdot (2 \hat{i} - λ \hat{j} + 3 \hat{k}) = 0$

We know that two planes $\vec{r} \cdot {\vec{n}}_{1} = d_{1}$ and $\vec{r} \cdot {\vec{n}}_{2} = d_{2}$ are perpendicular if

${\vec{n}}_{1} \cdot {\vec{n}}_{2} = 0$

Therefore,

$(λ \hat{i} + 5 \hat{j} - \hat{k}) \cdot (2 \hat{i} - λ \hat{j} + 3 \hat{k}) = 0$

$\Rightarrow 2 λ - 5 λ - 3 = 0$

$\Rightarrow - 3 λ - 3 = 0$

$\Rightarrow λ = - 1$

Therefore, $λ^{2} + λ = 0$

Stream:BITSATSubject:MathematicsTopic:3D-Coordinate GeometrySubtopic:Miscellaneous/Mixed

⏱ 2mℹ️ Source: PYQ_2018

Doubts & Discussion

Loading discussions...