Let a=2 i+7 j-k, b=3 i+5 k and c=i-j+2 k Let d be a vector which … - Vidyadip

MCQ

Let $\vec{a}=2 \hat{i}+7 \hat{j}-\hat{k}, \vec{b}=3 \hat{i}+5 \hat{k}$ and $\vec{c}=\hat{i}-\hat{j}+2 \hat{k}$ Let $\vec{d}$ be a vector which is perpendicular to both $\overrightarrow{ a }$ and $\overrightarrow{ b }, \quad$ and $\quad \overrightarrow{ c } \cdot \overrightarrow{ d }=12$. Then $(-\hat{i}+\hat{j}-\hat{k}) \cdot(\vec{c} \times \vec{d})$ is equal to $........$.

A
$48$
B
$42$
✓
$44$
D
$24$

✓

Answer

Correct option: C.

$44$

c
$\overrightarrow{ a }=2 \hat{ i }+7 \hat{ j }-\hat{ k }$

$\overrightarrow{ b }=3 \hat{ i }+5 \hat{ k }$

$\overrightarrow{ c }=\hat{ i }-\hat{ j }+2 \hat{ k }$

$\overrightarrow{ d }=\lambda(\overrightarrow{ a } \times \overrightarrow{ b })=\lambda\left|\begin{array}{ccc}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & 7 & -1 \\ 3 & 0 & 5\end{array}\right|$

$\overrightarrow{ d }=\lambda(35 \hat{ i }-13 \hat{ j }-21 \hat{ k })$

$\lambda(35+13-42)=12$

$\lambda=2$

$\overrightarrow{ d }=2(35 \hat{ i }-13 \hat{ j }-21 \hat{ k })$

$(\hat{ i }+\hat{ j }-\hat{ k })(\overrightarrow{ c } \times \overrightarrow{ d })$

$=\left|\begin{array}{ccc}-1 & 1 & -1 \\ 1 & -1 & 2 \\ 70 & -26 & -42\end{array}\right|=44$

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

More "M.C.Q (1 Marks)" from 10. vector algebra→10. vector algebra — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

If $\vec{a},\vec{b},\vec{c}$ are non-coplanar vectors and $(\vec{a} - \lambda \vec{b}) . (\vec{b} - 2\vec{c}) \times (\vec{c} + 2 \vec{a}) =0$ ,then $\lambda$ is equal to

A person writes 4 letters and addresses 4 envelopes. If the letters are placed in the envelopes at random, then the probability that all letters are not placed in the right envelopes, is

The value of $\int_0^1 {\frac{{dx}}{{{e^x} + {e^{ - x}}}}} $ is

Let $f : R \to R$ be differentiable at $c \in R$ and $f(c) = 0$. If $g\left( x \right) = \left| {f\left( x \right)} \right|$ , then at $x =c, g$ is

Let $R$ be a relation on $N$ defined by $x + 2y = 8$. The domain of $R$ is

The sum of the abosolute maximum and minimum values of the function $f(x)=\left|x^2-5 x+6\right|-3 x+2$ in the interval $[-1,3]$ is equal to :

If $A = \left[ {\begin{array}{*{20}{c}}1&0\\1&1\end{array}} \right]$ and $I = \left[ {\begin{array}{*{20}{c}}1&0\\0&1\end{array}} \right]$, then which one of the following holds for all $n \ge 1$, (by the principal of mathematical induction)

If $A=\left[\begin{array}{ll}\alpha & 2 \\ 2 & \alpha\end{array}\right]$ and $\left|A^3\right|=27$, then the value of $\alpha$ is

If $|\vec{\text{a}}|=\big|\vec{\text{b}}\big|,$ then $\big(\vec{\text{a}}+\vec{\text{b}}\big).\big(\vec{\text{a}}-\vec{\text{b}}\big)=$

Positive
Negetive
0
None of these

If $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}-\hat{\text{k}},\vec{\text{b}}=-\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}}$ and $\vec{\text{c}}=-\hat{\text{i}}+2\hat{\text{j}}-\hat{\text{k}},$ then a unit vector normal to the vectors $\vec{\text{a}}+\vec{\text{b}}$ and $\vec{\text{b}}-\vec{\text{c}}$ is:

$\hat{\text{i}}$
$\hat{\text{j}}$
$\hat{\text{k}}$
$\text{None of these}$