Let three vectors a- b and c be such that c is coplanar with a and b- a-c-7 and b is perpendicular to c where a-k- and b-2-k- then the value of 2-a-b-c-2 is -

Step 1: Finding c Given the three vectors a, b and c.Since c is coplane with a and b.⇒[abc]=0Also, it is given that b is perpendicular to c.⇒b·c=0Let c=λ(b×(a×b ...

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Byju's Answer

Standard XII

Mathematics

Slope of a Line

Let three vec...

Question

Let three vectors $\vec{a}$ , $\vec{b}$ and $\vec{c}$ be such that $\vec{c}$ is coplanar with $\vec{a}$ and $\vec{b}$ , $\vec{a} \cdot \vec{c} = 7$ and $\vec{b}$ is perpendicular to $\vec{c}$ where $\vec{a} = - \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = 2 \hat{i} + \hat{k}$ then the value of $2 {|\vec{a} + \vec{b} + \vec{c}|}^{2}$ is _______.

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Solution

Step 1: Finding $\vec{c}$
Given the three vectors $\vec{a}$ , $\vec{b}$ and $\vec{c}$ .
Since $\vec{c}$ is coplane with $\vec{a}$ and $\vec{b}$ .
$\Rightarrow [\vec{a} \vec{b} \vec{c}] = 0$
Also, it is given that $\vec{b}$ is perpendicular to $\vec{c}$ .
$\Rightarrow \vec{b} \cdot \vec{c} = 0$
Let $\vec{c} = λ (\vec{b} \times (\vec{a} \times \vec{b}))$
$\begin{array}{rcl} \Rightarrow & \vec{c} = λ [(\vec{b} \cdot \vec{b}) \vec{a} - (\vec{a} \cdot \vec{b}) \vec{b}] \\ = & λ [{|\vec{b}|}^{2} \vec{a} - (\vec{a} \cdot \vec{b}) \vec{b}] \end{array}$
Find the value of $|\vec{b}|$ using the given $\vec{b}$ .
$\begin{array}{rcl} {|\vec{b}|}^{2} & = & \sqrt{4 + 0 + 1} \\ = & \sqrt{5} \end{array}$
And the dot product of $\vec{a}$ and $\vec{b}$ .
$\Rightarrow \vec{a} \cdot \vec{b} = - 1$
Now substituting all the obtained values in $\begin{array}{rcl} \vec{c} & = & λ [{|\vec{b}|}^{2} \vec{a} - (\vec{a} \cdot \vec{b}) \vec{b}] \end{array}$ .
$\begin{array}{rcl} \vec{c} & = & λ (5 \cdot (- \hat{i} + \hat{j} + \hat{k}) - (- 1) (2 \hat{i} + \hat{k})) \\ = & λ (- 5 \hat{i} + 5 \hat{j} + 5 k + 2 \hat{i} + \hat{k}) \\ = & λ (- 3 \hat{i} + 5 \hat{j} + 6 \hat{k}) \end{array}$
Step 2: Finding the value of $λ$
We have given that $\vec{a} \cdot \vec{c} = 7$ using the condition we find the value $λ$ .
$\begin{array}{rcl} \vec{a} \cdot \vec{c} & = & 7 \\ (- \hat{i} + \hat{j} + \hat{k}) \cdot λ (- 3 \hat{i} + 5 \hat{j} + 6 \hat{k}) & = & 7 \\ λ (3 + 5 + 6) & = & 7 \\ 14 λ & = & 7 \\ λ & = & \frac{1}{2} \end{array}$
Step 3: Finding the value of $2 {|\vec{a} + \vec{b} + \vec{c}|}^{2}$
Substitute the value of $λ$ as $\frac{1}{2}$ in $\begin{array}{rcl} \vec{c} & = & λ (- 3 \hat{i} + 5 \hat{j} + 6 \hat{k}) \end{array}$ .
$\begin{array}{rcl} \vec{c} & = & \frac{1}{2} (- 3 \hat{i} + 5 \hat{j} + 6 \hat{k}) \\ = & - \frac{3}{2} \hat{i} + \frac{5}{2} \hat{j} + 3 \hat{k} \end{array}$
Now find the value of $|\vec{a} + \vec{b} + \vec{c}|$
$\begin{array}{rcl} |\vec{a} + \vec{b} + \vec{c}| & = & (- \hat{i} + \hat{j} + \hat{k}) + (2 \hat{i} + \hat{k}) + (- \frac{3}{2} \hat{i} + \frac{5}{2} \hat{j} + 3 \hat{k}) \\ = & - \hat{i} + \hat{j} + \hat{k} + 2 \hat{i} + \hat{k} - \frac{3}{2} \hat{i} + \frac{5}{2} \hat{j} + 3 \hat{k} \\ = & - \frac{1}{2} \hat{i} + \frac{7}{2} \hat{j} + 5 \hat{k} \end{array}$
Using the above result find $2 {|\vec{a} + \vec{b} + \vec{c}|}^{2}$
$\begin{array}{rcl} 2 {|\vec{a} + \vec{b} + \vec{c}|}^{2} & = & 2 {[\sqrt{{(\frac{1}{2})}^{2} + {(\frac{7}{2})}^{2} + 5^{2}}]}^{2} \\ = & 2 {[\sqrt{\frac{1}{4} + \frac{49}{4} + 25}]}^{2} \\ = & 2 (\frac{1 + 49 + 100}{4}) \\ = & 2 (\frac{150}{4}) \\ = & 75 \end{array}$
Therefore, the value of $2 {|\vec{a} + \vec{b} + \vec{c}|}^{2}$ is $75$ .

Suggest Corrections

Let three vectors a→, b→ and c→ be such that c→ is coplanar with a→ and b→, a→·c→=7 and b→ is perpendicular to c→ where a→=-i^+j^+k^ and b→=2i^+k^ then the value of 2a→+b→+c→2 is _______.