Let Z, a= i+j-k and b=3 i-j+2 k. Let c be a vector such that (a+b… - Vidyadip

Question

Let $\lambda \in Z, \vec{a}=\lambda \hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=3 \hat{i}-\hat{j}+2 \hat{k}$. Let $\overrightarrow{ c }$ be a vector such that $(\vec{a}+\vec{b}+\vec{c}) \times \vec{c}=\overrightarrow{0}, \vec{a} \cdot \vec{c}=-17$ and $\vec{b} \cdot \vec{c}=-20$ Then $|\overrightarrow{ c } \times(\lambda \hat{i}+\hat{j}+\hat{ k })|^2$ is equal to

✓

Answer

b
$(\vec{a}+\vec{b}+\overrightarrow{ c }) \times \overrightarrow{ c }=0$

$(\vec{a}+\vec{b}) \times \vec{c}=0$

$\overrightarrow{ c }=\alpha(\overrightarrow{ a }+\overrightarrow{ b })=\alpha(\lambda+3) \hat{ i }+\alpha \hat{ k }$

$\vec{b} \cdot \vec{c}=-20 \Rightarrow 3 \alpha(\lambda+3)+2 \alpha=-20$

$\vec{a} . \vec{c}=-17 \Rightarrow \alpha \lambda(\lambda+3)-\alpha=-17$

$\Rightarrow \alpha(3 \lambda+9+2)=-20$

$\alpha\left(\lambda^2+3 \lambda-1\right)=-17$

$17(3 \lambda+11)=20\left(\lambda^2+3 \lambda-1\right)$

$20 \lambda^2+9 \lambda-207=0$

$\lambda=3 \quad(\lambda \in Z)$

$\Rightarrow \alpha=-1 \quad \Rightarrow \overrightarrow{ c }=-(6 \hat{ i }+\hat{ k })$

$\overrightarrow{ v }=\overrightarrow{ c } \times(3 \hat{ i }+\hat{ j }+\hat{ k })$

$=\left|\begin{array}{lll}\hat{i} & \hat{j} & \hat{k} \\ -6 & 0 & -1 \\ 3 & 1 & 1\end{array}\right|=\hat{i}+3 \hat{j}-6 \hat{k}$

$|\overrightarrow{ v }|^2=(-1)^2+3^2+6^2=46$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

JEE STD 11 Science question papers→Gujarat Board STD 11 Science question papers→Gujarat Board question paper generator→

Similar questions

The number of four digit numbers that can be made with the digits $1,2,3,4$ and $5$ in which at least two digits are identical is

In a plane there are $37$ straight lines of which $13$ pass through the point $A$ and $11$ pass through the point $B$. Besides no three lines pass through one point, no line passes through both points $A$ and $B$ and no two are parallel. Then the number of intersection points the lines have is equal to

If $\frac{1}{2 \cdot 3^{10}}+\frac{1}{2^{2} \cdot 3^{9}}+\ldots \frac{1}{2^{10} \cdot 3}=\frac{K}{2^{10} \cdot 3^{10}}$, then the remainder when $K$ is divided by $6$ is

Let $\mathrm{P}(\alpha, \beta, \gamma)$ be the image of the point $\mathrm{Q}(1,6,4)$ in the line $\frac{x}{1}=\frac{y-1}{2}=\frac{z-2}{3}$. Then $2 \alpha+\beta+\gamma$ is equal to ..............

If sum of the first $21$ terms of the series $\log _{9^{1 / 2}} x +\log _{9^{1 / 3}} x +\log _{9^{1 / 4}} x +\ldots ., x >0$ , where $x>0$ is $504,$ then $\mathrm{x}$ is equal to:

Let $I(x)=\int \sqrt{\frac{x+7}{x}} d x$ and $I(9)=12+7 \log _e 7$ If I $(1)=\alpha+7 \log _e(1+2 \sqrt{2})$, then $\alpha^4$ is equal to $..........$.

Let $a_1, a_2, a_3, \ldots \ldots$ be an A.P. If $a_7=3$, the product $a_1 a_4$ is minimum and the sum of its first $n$ terms is zero, then $n !-4 a_{n(n+2)}$ is equal to :

The arithmetic mean and the geometric mean of two distinct 2-digit numbers $x$ and $y$ are two integers one of which can be obtained by reversing the digits of the other (in base 10 representation). Then, $x+y$ equals

The ordinates of the points $P$ and $Q$ on the parabola with focus $(3,0)$ and directrix $x =-3$ are in the ratio $3: 1$. If $R (\alpha, \beta)$ is the point of intersection of the tangents to the parabola at $P$ and $Q$, then $\frac{\beta^2}{\alpha}$ is equal to $.............$.

Tangents are drawn at points $(x_i,y_i); i = 1,2,3$ lying on $y = x^2$ to enclose a triangle of area $\Delta$.If $x_1,x_2,x_3$ form an increasing $AP$, where $x_1 =-1$ and $y_3 = 9$, then $\Delta$ is .............. $\mathrm{sq. \,units}$