Let a= i - k , b=x i + j +(1-x) k and c=y i +x j +(1+x-y) k . Then, [a b c] depends on

Correct answer: D. neither x nor y Here, [a b c]=|a ·(b × c)| Now, according to question, aligned [ array lll a & b c array ] & =| array ccc 1 & 0 & -1 x & 1 & 1 - x y & x & 1+x-y array | & =1(1+x-y-x(1-x))-1 (x^2-y) & =1+x-y-x+x^2-x^2+y & =1 & = Independent of x and y. aligned

BITSAT 2024

Exam: BITSAT
Year: 2024
Subject: Mathematics
Chapter: Vector Algebra

BITSAT 2024 · Mathematics · Vector Algebra

Let a= i - k , b=x i + j +(1-x) k and c=y i +x j +(1+x-y) k . Then,...

Let

\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{k}}, \mathbf{b}=x \hat{\mathbf{i}}+\hat{\mathbf{j}}+(1-x) \hat{\mathbf{k}}

and

\mathbf{c}=y \hat{\mathbf{i}}+x \hat{\mathbf{j}}+(1+x-y) \hat{\mathbf{k}}

. Then,

[\mathbf{a} \mathbf{b} \mathbf{ c}]

depends on

Options

Aonly $y$
Bonly $x$
Cboth $x$ and $y$
Dneither $x$ nor $y$

Correct answer

D. neither $x$ nor $y$

Step-by-step solution

Here,

[a b c]=|a \cdot(b \times c)|

Now, according to question, \begin{aligned} {\left[\begin{\text{ array }}{\text{ lll }} a & b c \end{\text{ array }}\right] } & =\left|\begin{array}{ccc} 1 & 0 & -1 \\ x & 1 & 1 - x \\ y & x & 1+x-y \end{array}\right| \\ & =1(1+x-y-x(1-x))-1 \left(x^2-y\right) \\ & =1+x-y-x+x^2-x^2+y \\ & =1 \\ & =\text{ Independent\text{ of } } x \text{\text{ and } } y. \end{aligned}

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