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Mathematics
Vector or cross product of two vectors
If $\overrightarrow{a}$ and $\overrightarrow{b}$ are two vectors, then \[\left( 2\overrightarrow{a}+3\overrightarrow{b} \right)\times \left( 5\overrightarrow{a}+7\overrightarrow{b} \right)+\left( \overrightarrow{a}\times \overrightarrow{b} \right)\] is equal to?
(a) $0$
(b) $1$
(c) \[\overrightarrow{a}\times \overrightarrow{b}\]
(d) \[\overrightarrow{b}\times \overrightarrow{a}\]

Mathematics
Vector or cross product of two vectors
If the vectors \[c\], \[a=xi+yj+zk\] and \[b=j\] are such that \[a\], \[c\] and \[b\] form a right handed system. Then \[c\] is?
(1) \[zi-xk\]
(2) \[0\]
(3) \[yj\]
(4) \[-zi+xk\]

Mathematics
Vector or cross product of two vectors
What is the vector product of two parallel vectors?

Mathematics
Vector or cross product of two vectors
Find a unit vector perpendicular to both the vectors \[\overrightarrow a \] and \[\overrightarrow b \], where \[\overrightarrow a = \hat i - 7\hat j + 7\hat k\] and $\overrightarrow b = 3\hat i - 2\hat j + 2\hat k$?
Mathematics
Vector or cross product of two vectors
If a and b are two non-zero non-collinear vectors then \[2\left[ {a{\text{ }}b{\text{ i}}} \right]{\text{i }} + {\text{ }}2\left[ {a{\text{ }}b{\text{ j}}} \right]{\text{j }} + {\text{ }}2\left[ {a{\text{ }}b{\text{ k}}} \right]{\text{k }} + \left[ {a{\text{ }}b{\text{ }}a} \right]\] is equal to ?
\[\left( 1 \right)\] \[2(a \times b)\]
\[\left( 2 \right)\] \[a \times b\]
\[\left( 3 \right)\] \[a + b\]
\[\left( 4 \right)\] None of these
Mathematics
Vector or cross product of two vectors
What does a cross product of $0$ mean?
Mathematics
Vector or cross product of two vectors
Let the position vectors of the points P and Q be \[4\overrightarrow i + \overrightarrow j + \lambda \overrightarrow k \] and \[2\overrightarrow i - \overrightarrow j + \lambda \overrightarrow k \] respectively. Vector \[\overrightarrow i - \overrightarrow j + 6\overrightarrow k \] is perpendicular to the plane containing the origin and the points P and Q. Then $\lambda $equals.
$A)\dfrac{{ - 1}}{2}$
$B)\dfrac{1}{2}$
$C)1$
$D) - 1$

Mathematics
Vector or cross product of two vectors
The value of \[\overset{\hat{\ }}{\mathop{i}}\,.\left( \overset{\hat{\ }}{\mathop{j}}\,\times \overset{\hat{\ }}{\mathop{k}}\, \right)+\overset{\hat{\ }}{\mathop{j}}\,.\left( \overset{\hat{\ }}{\mathop{i}}\,\times \overset{\hat{\ }}{\mathop{k}}\, \right)+\overset{\hat{\ }}{\mathop{k}}\,.\left( \overset{\hat{\ }}{\mathop{i}}\,\times \overset{\hat{\ }}{\mathop{j}}\, \right)\]
(A) 0
(B) -1
(C) 1
(D) 3
Mathematics
Vector or cross product of two vectors
If $\overrightarrow a ,\overrightarrow b ,\overrightarrow c $ are unit vectors such that $\overrightarrow a .\overrightarrow b = 0 = \overrightarrow a .\overrightarrow c $ and the angle between \[\overrightarrow b \]and \[\overrightarrow c \]is \[\dfrac{\pi }{3}\], then find the value of \[\left| {\overrightarrow a \times \overrightarrow b - \overrightarrow a \times \overrightarrow c } \right|\].
Mathematics
Vector or cross product of two vectors
Find the value of $\lambda \And \mu $ if $\left( 2\widehat{i}+6\widehat{j}+27\widehat{k} \right)\times \left( \widehat{i}+\lambda \widehat{j}+\mu \widehat{k} \right)=\overrightarrow{0}$?
Mathematics
Vector or cross product of two vectors
Can a cross product be negative?

Mathematics
Vector or cross product of two vectors
If ${{v}_{1}},{{v}_{2}},{{v}_{3}}$ are unit vectors given by
${{v}_{1}}=ai+bj+ck$
${{v}_{2}}=bi+cj+ak$
${{v}_{3}}=ci+aj+bk$
Where $a,b,c$ are non negative real numbers, and ${{v}_{\alpha }}.{{v}_{\beta }}=0$ , for $\alpha \ne \beta $ , then
(A) $\left| \left[ {{v}_{1}},{{v}_{2}},{{v}_{3}} \right] \right|=1$
(B) $a+b+c=1$
(C) ${{v}_{1}}+{{v}_{2}}+{{v}_{3}}=0$
(D) ${{v}_{1}},{{v}_{2}},{{v}_{3}}\text{ are coplanar}\text{.}$
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