JEE Advance - Mathematics (2021 - Paper 1 Online - No. 19)
Let $$\overrightarrow u $$, $$\overrightarrow v $$ and $$\overrightarrow w $$ be vectors in three-dimensional space, where $$\overrightarrow u $$ and $$\overrightarrow v $$ are unit vectors which are not perpendicular to each other and $$\overrightarrow u $$ . $$\overrightarrow w $$ = 1, $$\overrightarrow v $$ . $$\overrightarrow w $$ = 1, $$\overrightarrow w $$ . $$\overrightarrow w $$ = 4
If the volume of the paralleopiped, whose adjacent sides are represented by the vectors, $$\overrightarrow u $$, $$\overrightarrow v $$ and $$\overrightarrow w $$, is $$\sqrt 2 $$, then the value of $$\left| {3\overrightarrow u + 5\overrightarrow v } \right|$$ is ___________.
If the volume of the paralleopiped, whose adjacent sides are represented by the vectors, $$\overrightarrow u $$, $$\overrightarrow v $$ and $$\overrightarrow w $$, is $$\sqrt 2 $$, then the value of $$\left| {3\overrightarrow u + 5\overrightarrow v } \right|$$ is ___________.
Answer
7
Explanation
Given: $|\vec{u}|=1,|\vec{v}|=1$
$\vec{u}$ is not perpendicular to $\vec{v}$ $\vec{u} \cdot \vec{v} \neq 0$
And $\vec{u} \cdot \vec{w}=1 \quad \vec{v} \cdot \vec{w}=1 \quad \vec{w} \cdot \vec{w}=4$
$$ \begin{aligned} & \Rightarrow \vec{w} \cdot \vec{w}=|\vec{w}|^2=4 \\\\ & \Rightarrow|\vec{w}|=2 \end{aligned} $$
Volume of parallelepiped with $\vec{u}, \vec{v}$ and $\vec{w}$ as its sides,
$$ \begin{aligned} & =[\vec{u} \vec{v} \vec{w}]=\sqrt{2} \\\\ & \text { Now, }[\vec{u} \vec{v} \vec{w}]^2=\left|\begin{array}{ccc} \vec{u} \cdot \vec{u} & \vec{u} \cdot \vec{v} & \vec{u} \cdot \vec{w} \\ \vec{v} \cdot \vec{u} & \vec{v} \cdot \vec{v} & \vec{v} \cdot \vec{w} \\ \vec{w} \cdot \vec{u} & \vec{w} \cdot \vec{v} & \vec{w} \cdot \vec{w} \end{array}\right|=2 \\\\ & \Rightarrow\left|\begin{array}{ccc} 1 & \vec{u} \cdot \vec{v} & 1 \\ \vec{v} \cdot \vec{u} & 1 & 1 \\ 1 & 1 & 4 \end{array}\right|=2 \end{aligned} $$
$$ \begin{aligned} & \Rightarrow 1(4-1)-\vec{u} \cdot \vec{v}(4 \vec{u} \cdot \vec{v}-1)+1(\vec{u} \cdot \vec{v}-1)=2 \\\\ & \Rightarrow 3-4(\vec{u} \vec{v})^2+\vec{u} \vec{v}+\vec{u} \vec{v}-1=2 \\\\ & \Rightarrow-4(\vec{u} \vec{v})^2+2 \vec{u} \vec{v}+2=2 \\\\ & \Rightarrow-4(\vec{u} \vec{v})^2+2 \vec{u} \vec{v}=0 \\\\ & \Rightarrow 2 \vec{u} \vec{v}(-2 \vec{u} \vec{v}+1)=0 \\\\ & \Rightarrow \vec{u} \cdot \vec{v}=\frac{1}{2}, \vec{u} \cdot \vec{v} \neq 0 \\\\ & \text { Now, }|3 \vec{u}+5 \vec{v}|=\sqrt{9+25+30\left(\frac{1}{2}\right)} \\\\ & \Rightarrow|3 \vec{u}+5 \vec{v}|=\sqrt{49} \\\\ & \Rightarrow|3 \vec{u}+5 \vec{v}|=7 \end{aligned} $$
$\vec{u}$ is not perpendicular to $\vec{v}$ $\vec{u} \cdot \vec{v} \neq 0$
And $\vec{u} \cdot \vec{w}=1 \quad \vec{v} \cdot \vec{w}=1 \quad \vec{w} \cdot \vec{w}=4$
$$ \begin{aligned} & \Rightarrow \vec{w} \cdot \vec{w}=|\vec{w}|^2=4 \\\\ & \Rightarrow|\vec{w}|=2 \end{aligned} $$
Volume of parallelepiped with $\vec{u}, \vec{v}$ and $\vec{w}$ as its sides,
$$ \begin{aligned} & =[\vec{u} \vec{v} \vec{w}]=\sqrt{2} \\\\ & \text { Now, }[\vec{u} \vec{v} \vec{w}]^2=\left|\begin{array}{ccc} \vec{u} \cdot \vec{u} & \vec{u} \cdot \vec{v} & \vec{u} \cdot \vec{w} \\ \vec{v} \cdot \vec{u} & \vec{v} \cdot \vec{v} & \vec{v} \cdot \vec{w} \\ \vec{w} \cdot \vec{u} & \vec{w} \cdot \vec{v} & \vec{w} \cdot \vec{w} \end{array}\right|=2 \\\\ & \Rightarrow\left|\begin{array}{ccc} 1 & \vec{u} \cdot \vec{v} & 1 \\ \vec{v} \cdot \vec{u} & 1 & 1 \\ 1 & 1 & 4 \end{array}\right|=2 \end{aligned} $$
$$ \begin{aligned} & \Rightarrow 1(4-1)-\vec{u} \cdot \vec{v}(4 \vec{u} \cdot \vec{v}-1)+1(\vec{u} \cdot \vec{v}-1)=2 \\\\ & \Rightarrow 3-4(\vec{u} \vec{v})^2+\vec{u} \vec{v}+\vec{u} \vec{v}-1=2 \\\\ & \Rightarrow-4(\vec{u} \vec{v})^2+2 \vec{u} \vec{v}+2=2 \\\\ & \Rightarrow-4(\vec{u} \vec{v})^2+2 \vec{u} \vec{v}=0 \\\\ & \Rightarrow 2 \vec{u} \vec{v}(-2 \vec{u} \vec{v}+1)=0 \\\\ & \Rightarrow \vec{u} \cdot \vec{v}=\frac{1}{2}, \vec{u} \cdot \vec{v} \neq 0 \\\\ & \text { Now, }|3 \vec{u}+5 \vec{v}|=\sqrt{9+25+30\left(\frac{1}{2}\right)} \\\\ & \Rightarrow|3 \vec{u}+5 \vec{v}|=\sqrt{49} \\\\ & \Rightarrow|3 \vec{u}+5 \vec{v}|=7 \end{aligned} $$
Comments (0)
