Let a-2-k- b-k-and c-k- A vector coplanar to a and b has a projection along c of magnitude 1-3- then the vector is-

Explanation for the correct options:Determine the required vector.Let r is the vector coplanar to a and b, then;r = a+ m br= (î+2ĵ+k̂) + m(î ĵ+k̂) ...

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Byju's Answer

Standard XII

Mathematics

Applications of Cross Product

Let a=î+2ĵ+k̂...

Question

Let $\vec{a} = \hat{i} + 2 \hat{j} + \hat{k}, \vec{b} = \hat{i} - \hat{j} + \hat{k}$ and $\vec{c} = \hat{i} + \hat{j} - \hat{k}$ . A vector coplanar to $\vec{a}$ and $\vec{b}$ has a projection along $\vec{c}$ of magnitude $\frac{1}{\sqrt{3}}$ , then the vector is?

$4 \hat{i} - \hat{j} + 4 \hat{k}$

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$4 \hat{i} + \hat{j} - 4 \hat{k}$

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$2 \hat{i} + \hat{j} + \hat{k}$

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None of these

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Solution

The correct option is A
$4 \hat{i} - \hat{j} + 4 \hat{k}$

Explanation for the correct options:
Determine the required vector.
Let $\vec{r}$ is the vector coplanar to $\vec{a}$ and $\vec{b}$ , then;
$\vec{r} = \vec{a} + m \vec{b}$
$\vec{r} = (\hat{i} + 2 \hat{j} + \hat{k}) + m (\hat{i} - \hat{j} + \hat{k}) [∵ \vec{a} = \hat{i} + 2 \hat{j} + \hat{k} & \vec{b} = \hat{i} - \hat{j} + \hat{k}] \vec{r} = \hat{i} (1 + m) + \hat{j} (2 - m) + \hat{k} (1 + m) . . . . . (i)$
Since the projection of $\vec{r}$ along $\vec{c}$ is $\frac{1}{\sqrt{3}}$ then
$\Rightarrow \frac{\vec{r} . \vec{c}}{| \vec{c} |} = \pm \frac{1}{\sqrt{3}} \Rightarrow \frac{(1 + m) \hat{i} . \hat{i} + (2 - m) \hat{j} . \hat{j} - (1 + m) \hat{k} . \hat{k}}{\sqrt{1^{2} + 1^{2} + 1^{2}}} = \pm \frac{1}{\sqrt{3}} [∵ c = \hat{i} + \hat{j} - \hat{k} a n d \hat{i}, \hat{j}, \hat{k} a r e u n i t v e c t o r s a n d \hat{i} . \hat{i} = \hat{j} . \hat{j} = \hat{k} . \hat{k} = 1] \Rightarrow (1 + m) + (2 - m) - (1 + m) = \pm 1 \Rightarrow (2 - m) = \pm 1 \Rightarrow m = 3 o r - 1$
Now putting $m = - 1$ into $(i)$ we get
$\Rightarrow \vec{r} = \hat{i} (1 - 1) + \hat{j} (2 + 1) + \hat{k} (1 - 1) \Rightarrow \vec{r} = 3 \hat{j}$
Again putting Now putting $m = 3$ into $(i)$ we get
$\Rightarrow \vec{r} = \hat{i} (1 + 3) + \overset{}{\hat{j}} (2 - 3) + \hat{k} (1 + 3) \Rightarrow \vec{r} = 4 \hat{i} - \hat{j} + 4 \hat{k}$
Hence, option A is the correct answer.

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Let a→=i^+2j^+k^,b→=i^-j^+k^and c→=i^+j^-k^. A vector coplanar to a→ and b→ has a projection along c→ of magnitude 13, then the vector is?

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