Let a vector a be coplanar with vectors b-2-k- and c-k- If a is perpendicular to d-3-2-6k- and -a-10- Then a possible value of -a b c-a b d-a c d- is equal to

Given: nbsp;b=2î+ĵ+k̂, c=î ĵ+k̂ d=3î+2ĵ+6k̂ |a|=√(10) Now, nbsp;[b c d] = | 2 amp; 1 amp;1 nbsp; 1 amp; 1 amp; 1 nbsp; 3 amp; 2 a ...

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Byju's Answer

Standard XII

Mathematics

Addition of Vectors

Let a vector ...

Question

Let a vector $\to a$ be coplanar with vectors $\to b = 2^i +^j +^k$ and $\to c =^i -^j +^k .$ If $\to a$ is perpendicular to $\to d = 3^i + 2^j + 6^k$ , and $| \to a | = \sqrt{10} .$ Then a possible value of $[\to a \to b \to c] + [\to a \to b \to d] + [\to a \to c \to d]$ is equal to

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Solution

The correct option is C $- 42$
Given: $\to b = 2^i +^j +^k, \to c =^i -^j +^k$
$\to d = 3^i + 2^j + 6^k$
$| \to a | = \sqrt{10}$
Now, $[\to b \to c \to d] = ∣ ∣ ∣ ∣ \begin{matrix} 2 & 1 & 1 1 & - 1 & 1 3 & 2 & 6 \end{matrix} ∣ ∣ ∣ ∣$
$= 2 (- 6 - 2) - 1 (3) + 1 (5) = - 14$
As $\to a$ is coplanar with vectors $\to b$ and $\to c$ , so assuming
$\to a = λ \to b + μ \to c \Rightarrow \to a = (2 λ + μ)^i + (λ - μ)^j + (λ + μ)^k$

We know, $\to a$ is perpendicular to $\to d$
$\Rightarrow \to a \cdot \to d = 0 \Rightarrow 3 (2 λ + μ) + 2 (λ - μ) + 6 (λ + μ) = 0 \Rightarrow 14 λ + + 7 μ = 0 \Rightarrow μ = - 2 λ \dots (1)$
Also, $| \to a | = \sqrt{10}$
$\Rightarrow (2 λ + μ)^{2} + (λ - μ)^{2} + (λ + μ)^{2} = 10$
From equation $(1)$ , we get
$\Rightarrow (2 λ - 2 λ)^{2} + (λ + 2 λ)^{2} + (λ - 2 λ)^{2} = 10 \Rightarrow 10 λ^{2} = 10 \Rightarrow λ = \pm 1 \Rightarrow μ = \mp 2$
Now,
$[\to a \to b \to c] + [\to a \to b \to d] + [\to a \to c \to d] = 0 + [\to a \to b \to d] + [\to a \to c \to d] = [(λ \to b + μ \to c) \to b \to d] + [(λ \to b + μ \to c) \to c \to d] = λ [\to b \to b \to d] + μ [\to c \to b \to d] + λ [\to b \to c \to d] + μ [\to c \to c \to d] = - μ [\to b \to c \to d] + λ [\to b \to c \to d] = (λ - μ) [\to b \to c \to d]$
From equation $(1)$ , we get
$= - 42 λ = \mp 42$

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Let a vector →a be coplanar with vectors →b=2^i+^j+^k and →c=^i−^j+^k. If →a is perpendicular to →d=3^i+2^j+6^k, and |→a|=√10. Then a possible value of [→a →b →c]+[→a →b →d]+[→a →c →d] is equal to

Let a vector $\to a$ be coplanar with vectors $\to b = 2^i +^j +^k$ and $\to c =^i -^j +^k .$ If $\to a$ is perpendicular to $\to d = 3^i + 2^j + 6^k$ , and $| \to a | = \sqrt{10} .$ Then a possible value of $[\to a \to b \to c] + [\to a \to b \to d] + [\to a \to c \to d]$ is equal to