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Vector Component

If u⃗=ĵ+4k̂...

Question

If $\to u =^j + 4^k$ , $\to v =^i - 3^k$ , and $\to w = cos θ^i + sin θ^j$ are vectors in $3$ -dimensional space, then the maximum possible value of $| \to u \times \to v \cdot \to w |$ is?

\sqrt{14}

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5

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7

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\sqrt{13}

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Solution

The correct option is B $5$
Given $\to u =^j + 4^k, \to v =^i - 3^k, ¯ ω = c o s θ^i + s i n θ^j$
$| \to u \times \to v . \to ω |$ $= ∣ ∣ ∣ ∣ \begin{matrix} 0 & 1 & 4 1 & 0 & - 3 c o s θ & s i n θ & 0 \end{matrix} ∣ ∣ ∣ ∣$
$- 1 (3 c o s θ) + 4 s i n θ$
$= 4 s i n θ - 3 c o s θ$
Let $f (θ) = 4 s i n θ - 3 c o s θ$
$f^{'} (θ) = 4 c o s θ + 3 s i n θ = 0$
$\Rightarrow t a n θ = \frac{- 4}{3}$
$s i n θ = \frac{4}{5}$ and $c o s θ = \frac{- 3}{5}$
$f (θ) = 4 \times \frac{4}{5} + 3 \times \frac{3}{5} = 5$
$∴$ The maximum value of $| \to u \times \to v . \to ω |$ is 5.

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Similar questions

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If →u=^j+4^k, →v=^i−3^k, and →w=cosθ^i+sinθ^j are vectors in 3-dimensional space, then the maximum possible value of |→u×→v⋅→w| is?

If $\to u =^j + 4^k$ , $\to v =^i - 3^k$ , and $\to w = cos θ^i + sin θ^j$ are vectors in $3$ -dimensional space, then the maximum possible value of $| \to u \times \to v \cdot \to w |$ is?