Let a → = 3 i ^ + 2 j ^ + x k ^ and b → = i ^ - j ^ + k ^ , for some real x . Then the condition for a → × b → = r to follow

As, We know that a → × b → = i ^ j ^ k ^ 3 2 x 1 - 1 1 = 2 + x i ^ + x - 3 j ^ - 5 k ^ a → × b → = 4 + x 2 + 4 x + x 2 + 9 - 6 x + 25 = 2 x 2 - 2 x + 38 Therefore, we have ⇒ a → × b → ≥ 75 2 ⇒ a → × b → ≥ 5 3 2

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MathematicsVectorDot Product & Its Application ( Projection etc.)Easy2 minPYQ_2019

MathematicsEasysingle choice

Let $\vec{a} = 3 \hat{i} + 2 \hat{j} + x \hat{k}$ and $\vec{b} = \hat{i} - \hat{j} + \hat{k},$ for some real $x$ . Then the condition for $|\vec{a} \times \vec{b}| = r$ to follow

Options:

Answer:

Solution:

As, We know that $\vec{a} \times \vec{b} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & 2 & x \\ 1 & - 1 & 1 \end{matrix}|$
$= (2 + x) \hat{i} + (x - 3) \hat{j} - 5 \hat{k}$

$|\vec{a} \times \vec{b}| = \sqrt{4 + x^{2} + 4 x + x^{2} + 9 - 6 x + 25}$

$= \sqrt{2 x^{2} - 2 x + 38}$

Therefore, we have
$\Rightarrow |\vec{a} \times \vec{b}| \geq \sqrt{\frac{75}{2}}$

$\Rightarrow |\vec{a} \times \vec{b}| \geq 5 \sqrt{\frac{3}{2}}$

Stream:JEESubject:MathematicsTopic:VectorSubtopic:Dot Product & Its Application ( Projection etc.)

⏱ 2mℹ️ Source: PYQ_2019

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