SATHEE: Vectors 5 Question 3

Vectors 5 Question 3

3. Let $\vec{a} = 2 \hat{i} + \hat{j} + \hat{k}, \vec{b} = \hat{i} + 2 \hat{j} - \hat{k}$ and a unit vector $\vec{c}$ be coplanar. If $\vec{c}$ is perpendicular to $\vec{a}$ , then $\vec{c}$ is equal to

$(1999, 2 M)$

(a) $\frac{1}{\sqrt{2}} (- \hat{j} + \hat{k})$

(b) $\frac{1}{\sqrt{3}} (- \hat{i} - \hat{j} - \hat{k})$

(d) $\frac{1}{\sqrt{5}} (\hat{i} - \hat{j} - \hat{k})$

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Answer:

Correct Answer: 3. (a)

Solution:

It is given that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$ , we take

$\vec{c} = p \vec{a} + q \vec{b}$

where, $p$ and $q$ are scalars.

Since, $\vec{c} ⊥ \vec{a} \Rightarrow \vec{c} \cdot \vec{a} = 0$

Taking dot product of $\vec{a}$ in Eq. (i), we get

$\begin{aligned} \vec{c} \cdot \vec{a} = p \vec{a} \cdot \vec{a} + q \vec{b} \cdot \vec{a} \Rightarrow 0 = p | \vec{a} |^{2} + q | \vec{b} \cdot \vec{a} | \\ [\begin{aligned} ∵ \vec{a} & = 2 \hat{i} + \hat{j} + \hat{k} \\ | \vec{a} | & = \sqrt{2^{2} + 1 + 1} = \sqrt{6} . \\ \vec{a} \cdot \vec{b} & = (2 \hat{i} + \hat{j} + \hat{k}) \cdot (\hat{i} + 2 \hat{j} - \hat{k}) \\ = 2 + 2 - 1 = 3 \end{aligned}] \\ \Rightarrow 0 = p \cdot 6 + q \cdot 3 \Rightarrow q = - 2 p \end{aligned}$

On putting in Eq. (i), we get

$\begin{aligned} \vec{c} = p \vec{a} + \vec{b} (- 2 p) \\ \Rightarrow & \vec{c} = p \vec{a} - 2 p \vec{b} \Rightarrow \vec{c} = p (\vec{a} - 2 \vec{b}) \\ \Rightarrow & \vec{c} = p [(2 \hat{i} + \hat{j} + \hat{k}) - 2 (\hat{i} + 2 \hat{j} - \hat{k})] \\ \Rightarrow & \vec{c} = p (- 3 \hat{j} + 3 \hat{k}) \Rightarrow | \vec{c} | = p \sqrt{(- 3)^{2} + 3^{2}} \\ \Rightarrow & | \vec{c} |^{2} = p^{2} (\sqrt{18})^{2} \Rightarrow | \vec{c} |^{2} = p^{2} \cdot 18 \\ \Rightarrow & 1 = p^{2} \cdot 18 \\ \Rightarrow & p^{2} = \frac{1}{18} \Rightarrow p = \pm \frac{1}{3 \sqrt{2}} ∣= 1] \\ ∴ & \vec{c} = \pm \frac{1 (- \hat{j} + \hat{k})}{\sqrt{2}} \end{aligned}$

Vectors 5 Question 3

3. Let $\vec{a} = 2 \hat{i} + \hat{j} + \hat{k}, \vec{b} = \hat{i} + 2 \hat{j} - \hat{k}$ and a unit vector $\vec{c}$ be coplanar. If $\vec{c}$ is perpendicular to $\vec{a}$ , then $\vec{c}$ is equal to

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Vectors 5 Question 3

3. Let a→=2i^+j^+k^,b→=i^+2j^−k^ and a unit vector c→ be coplanar. If c→ is perpendicular to a→, then c→ is equal to

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3. Let $\vec{a} = 2 \hat{i} + \hat{j} + \hat{k}, \vec{b} = \hat{i} + 2 \hat{j} - \hat{k}$ and a unit vector $\vec{c}$ be coplanar. If $\vec{c}$ is perpendicular to $\vec{a}$ , then $\vec{c}$ is equal to