If a , b , c are three non-coplanar vectors, then ( a + b + c ). … - Vidyadip

Question

If $\vec{\text{a}},\vec{\text{b}},\vec{\text{c}}$ are three non-coplanar vectors, then $\big(\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}\big).\big[\big(\vec{\text{a}}+\vec{\text{b}}\big)\times\big(\vec{\text{a}}+\vec{\text{c}}\big)\big]$ equals:

$0$
$\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]$
$2\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]$
$-\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]$

✓

Answer

$-\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]$

Solution:

$\big(\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}\big).\big[\big(\vec{\text{a}}+\vec{\text{b}}\big)\times\big(\vec{\text{a}}+\vec{\text{c}}\big)\big]$

$=\big(\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}\big).\big(\vec{\text{a}}\times\vec{\text{a}}+\vec{\text{a}}\times\vec{\text{c}}+\vec{\text{b}}\times\vec{\text{a}}+\vec{\text{b}}\times\vec{\text{c}}\big)$

$=\big(\vec{\text{a}}+\vec{\text{b}}+\vec{\text{c}}\big).\big(\vec{\text{a}}\times\vec{\text{c}}+\vec{\text{b}}\times\vec{\text{a}}+\vec{\text{b}}\times\vec{\text{c}}\big)$

$=0+0+\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]+\big[\vec{\text{b}}\vec{\text{a}}\vec{\text{c}}\big]+0+0+0+\big[\vec{\text{c}}\vec{\text{b}}\vec{\text{a}}\big]+0$

$=-\big[\vec{\text{a}}\vec{\text{b}}\vec{\text{c}}\big]$

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 VECTOR ALGEBRA→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

Let $f: R \rightarrow R$ be defined as $f(x)=x-1$ and $g: R -\{1,-1\} \rightarrow R$ be defined as $g(x)=\frac{x^{2}}{x^{2}-1}$. Then the function fog is

If $\text{A}=\begin{bmatrix}\text{i}&0\\0&\text{i}\end{bmatrix},\text{n}\in\text{N},$ then A⁴ⁿ equals:

$\begin{bmatrix}0&\text{i}\\\text{i}&0\end{bmatrix}$
$\begin{bmatrix}0&0\\0&0\end{bmatrix}$
$\begin{bmatrix}1&0\\0&1\end{bmatrix}$
$\begin{bmatrix}0&\text{i}\\\text{i}&0\end{bmatrix}$

A matrix consisting of a single column of m elements is know as:

Column matrix
Row matrix
Square matrix
Null matrix

If a dice is thrown twice, the probability of occurrence of $4$ at least once is

Let a function $f\left( x \right) = \left\{ \begin{gathered}
- \ln \left( {3x - \left[ {3x} \right]} \right)\,;\,\,3x \ne n;n \in N \hfill \\
\ln \left( {\operatorname{sgn} \left( {3x} \right)} \right)\,\,\,\,\,\,\,;\,\,3x = n;n \in N \hfill \\
\end{gathered} \right.,$ (where [.] and sgn $(x)$ denotes greatest integer function and signum function respectively) then number of point $(s)$ , where $f(x)$ is minimum in $x \in (0, 5)$ , is

Find the value of x, if $\begin{bmatrix}2&5\\3&\text{x}\end{bmatrix}=\begin{bmatrix}\text{x}&-1\\5&3\end{bmatrix}$ is:

20
-20
30
-30

If two events are independent, then.

The number of arbitrary constants in the particular solution of a differential equation of third order is:

3
2
1
0

The graph of the function $f(x)=x+\frac{1}{8} \sin (2 \pi x), 0 \leq x \leq 1$ is shown below. Define $f_1(x)=f(x), f_{n+1}(x)=f\left(f_n(x)\right)$, for $n \geq 1$.

Which of the following statements are true?

$I.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)=0$

$II.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)=\frac{1}{2}$

$III.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)=1$

$IV.$ There are infinitely many $x \in[0,1]$ for which $\lim _{n \rightarrow \infty} f_n(x)$ does not exist.

The moment of a force represented by $\overrightarrow F = i + 2j + 3k$ about the point $2\,i - j + k = $