Vector Algebra · Mathematics · WB JEE

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and $\lambda$ is a real number then the vectors $\vec{a}+2 \vec{b}+3 \vec{c}, \lambda \vec{b}+4 \vec{c}$ and $(2 \lambda-1) \vec{c}$ are non-coplanar for

If ' $\theta$ ' is the angle between two vectors $\vec{a}$ and $\vec{b}$ such that $|\vec{a}|=7,|\vec{b}|=1$ and $|\vec{a} \times \vec{b}|^2=k^2-(\vec{a} \cdot \vec{b})^2$, then the values of $k$ and $\theta$ are

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be vectors of equal magnitude such that the angle between $\vec{a}$ and $\vec{b}$ is $\alpha, \vec{b}$ and $\vec{c}$ is $\beta$ and $\vec{c}$ and $\vec{a}$ is $\gamma$. Then the minimum value of $\cos \alpha+\cos \beta+\cos \gamma$ is

If $\vec{\alpha}=3 \vec{i}-\vec{k},|\vec{\beta}|=\sqrt{5}$ and $\vec{\alpha} \cdot \vec{\beta}=3$, then the area of the parallelogram for which $\vec{\alpha}$ and $\vec{\beta}$ are adjacent sides is

Let $\vec{a}, \vec{b}, \vec{c}$ be unit vectors. Suppose $\vec{a} \cdot \vec{b}=\vec{a} \cdot \vec{c}=0$ and the angle between $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{6}$. Then $\vec{a}$ is

A unit vector in XY-plane making an angle $$45^{\circ}$$ with $$\hat{i}+\hat{j}$$ and an angle $$60^{\circ}$$ with $$3 \hat{i}-4 \hat{j}$$ is

The value of 'a' for which the scalar triple product formed by the vectors $$\overrightarrow \alpha = \widehat i + a\widehat j + \widehat k,\overrightarrow \beta = \widehat j + a\widehat k$$ and $$\overrightarrow \gamma = a\widehat i + \widehat k$$ is maximum, is

If the volume of the parallelopiped with $$\overrightarrow a \times \overrightarrow b ,\overrightarrow b \times \overrightarrow c $$ and $$\overrightarrow c \times \overrightarrow a $$ as conterminous edges is 9 cu. units, then the volume of the parallelopiped with $$(\overrightarrow a \times \overrightarrow b ) \times (\overrightarrow b \times \overrightarrow c ),(\overrightarrow b \times \overrightarrow c ) \times (\overrightarrow c \times \overrightarrow a )$$, and $$(\overrightarrow c \times \overrightarrow a ) \times (\overrightarrow a \times \overrightarrow b )$$ as conterminous edges is

If $$\overrightarrow a = \widehat i + \widehat j - \widehat k$$, $$\overrightarrow b = \widehat i - \widehat j + \widehat k$$ and $$\overrightarrow c $$ is unit vector perpendicular to $$\overrightarrow a $$ and coplanar with $$\overrightarrow a $$ and $$\overrightarrow b $$, then unit vector $$\overrightarrow d $$ perpendicular to both $$\overrightarrow a $$ and $$\overrightarrow c $$ is

If $${\overrightarrow \alpha }$$ is a unit vector, $$\overrightarrow \beta = \widehat i + \widehat j - \widehat k$$, $$\overrightarrow \gamma = \widehat i + \widehat k$$ then the maximum value of $$\left[ {\overrightarrow \alpha \overrightarrow \beta \overrightarrow \gamma } \right]$$ is