Let a three-dimensional vector V satisfy the condition 2V-V-2-2-k- If 3-V-m- then the value of m is

2V+V×(î+2ĵ)=2î+k̂ 2 V· (î+2ĵ)+0=(2î+k̂)· (î+2ĵ) 2V·(î+2ĵ)=2 |V·(î+2ĵ)|^2=1 |V|^2·|î+2ĵ|^2cos^2θ=1 nbsp; nbsp; nbsp; nbsp;θ is the angle between V an ...

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Standard XII

Mathematics

Applications of Cross Product

Let a three-d...

Question

Let a three-dimensional vector $\to V$ satisfy the condition $2 \to V + \to V \times (^i + 2^j) = 2^i +^k$ . If $3 | \to V | = \sqrt{m}$ , then the value of $m$ is

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Solution

$2 \to V + \to V \times (^i + 2^j) = 2^i +^k$
$2 \to V \cdot (^i + 2^j) + 0 = (2^i +^k) \cdot (^i + 2^j)$
$2 \to V \cdot (^i + 2^j) = 2$
$| \to V \cdot (^i + 2^j) |^{2} = 1$
$| \to V |^{2} \cdot |^i + 2^j |^{2} {cos}^{2} θ = 1$
$θ$ is the angle between $\to V$ and $^i + 2^j$
$| \to V |^{2} 5 (1 - {sin}^{2} θ) = 1$
$| \to V |^{2} 5 {sin}^{2} θ = 5 | \to V |^{2} - 1 (i i)$
From eq $(i)$ , we have
$| 2 \to V + \to V \times (^i + 2^j) |^{2} = | 2^i +^k |^{2}$
$4 | \to V |^{2} + | \to V \times (^i + 2^j) |^{2} = 5$
$4 | \to V |^{2} + | \to V |^{2} \cdot |^i + 2^j |^{2} {sin}^{2} θ = 5$
$4 | \to V |^{2} + 5 | \to V |^{2} {sin}^{2} θ = 5$
$4 | \to V |^{2} + 5 | \to V |^{2} - 1 = 5$
$9 | \to V |^{2} = 6$
$3 | \to V | = \sqrt{6} = \sqrt{m}$
$\Rightarrow m = 6$

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Let a three-dimensional vector →V satisfy the condition 2→V+→V×(^i+2^j)=2^i+^k. If 3|→V|=√m, then the value of m is

Let a three-dimensional vector $\to V$ satisfy the condition $2 \to V + \to V \times (^i + 2^j) = 2^i +^k$ . If $3 | \to V | = \sqrt{m}$ , then the value of $m$ is