If vectors- a-2-k- b-2-3k-c-3-5k- are coplanar- Then the value of - is equal to-

Explanation for correct option:Finding the value of λ:Given vectors a, b,c are co planersthen,|abc| = 0 or |[ 2 1 1; 1 2 3; 3 λ 5 ]|=02(10+3λ) ( 1)(5+9 ...

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Standard XII

Mathematics

Test for Collinearity of Vectors

If vectors, a...

Question

If vectors, $\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}$ , $\vec{b} = \hat{i} + 2 \hat{j} - 3 \hat{k}$ , $\vec{c} = 3 \hat{i} + λ \hat{j} + 5 \hat{k}$ are coplanar. Then the value of $λ$ is equal to?

$λ = 4$

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$λ = - 4$

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$λ = - 7$

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$λ = 7$

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Solution

The correct option is B
$λ = - 4$

Explanation for correct option:
Finding the value of $λ$ :
Given vectors $\vec{a}$ , $\vec{b}$ , $\vec{c}$ are co planers
then, $| a b c | = 0$ or $|\begin{matrix} 2 & - 1 & 1 \\ 1 & 2 & - 3 \\ 3 & λ & 5 \end{matrix}| = 0$
$2 (10 + 3 λ) - (- 1) (5 + 9) + 1 (λ - 6) = 0 [Expanding along row 1] \Rightarrow 7 λ + 28 = 0 \Rightarrow λ = - 4$
Hence, the correct option is C.

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Similar questions

Q. The value of

λ

, for which the four points

^i -^j +^k, 2^i + 2^j - 3^k, 3^i + 2^j - 2^k,^i + λ^j + 2^k

are coplanar, is:

Q. Find the value of

λ

are the vectors

λ^i + 2^j +^k

3^i + 2^j + λ^k,^i +^j

are coplanar.

Q. Let

\to a = 2^i -^j +^k, \to b =^i + 2^j -^k, \to c =^i +^j - 2^k

be three vectors. A vector in the plane of

\to b

and

\to c

whose projection on

\to a

is of magnitude

\sqrt{\frac{2}{3}}

Q. Let

x_{0}

be the point of Local maxima of

f (x) =

\to a \cdot (\to b \times \to c),

where

\to a = x^i - 2^j + 3^k, \to b = - 2^i + x^j -^k

and

\to c = 7^i - 2^j + x^k

. Then the value of

\to a \cdot \to b + \to b \cdot \to c + \to c \cdot \to a

x = x_{0}

Q. Let

\to u

be a vector coplanar with the vectors

\to a = 2^i + 3^j -^k

and

\to b =^j +^k

. If

\to u

is perpendicular to

\to a

and

\to u \cdot \to b = 24,

then

| \to u |^{2}

is equal to

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If vectors, a→=2i^−j^+k^, b→=i^+2j^−3k^,c→=3i^+λj^+5k^ are coplanar. Then the value of λ is equal to?

The correct option is B λ=-4Explanation for correct option:Finding the value of λ:Given vectors a→, b→,c→ are co planersthen,|abc|=0 or 2-1112-33λ5=0210+3λ--15+9+1λ-6=0Expandingalongrow1⇒7λ+28=0⇒λ=-4Hence, the correct option is C.

If vectors, $\vec{a} = 2 \hat{i} - \hat{j} + \hat{k}$ , $\vec{b} = \hat{i} + 2 \hat{j} - 3 \hat{k}$ , $\vec{c} = 3 \hat{i} + λ \hat{j} + 5 \hat{k}$ are coplanar. Then the value of $λ$ is equal to?