Vector Algebra - Algebra - Mathematics Previous Year Questions

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

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

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

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

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

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 $\vec{\alpha}=3 \vec{i}-\vec{k},|\vec{\beta}|=\sqrt{5}$ 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 $\vec{a}, \vec{b}$ 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

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

For any four vectors $\mathbf{a , b , c , d}$ the expression $(\mathrm{b} \times \mathrm{c}) \cdot(\mathrm{a} \times \mathrm{d})+(\mathrm{c} \times \mathrm{a}) \cdot(\mathrm{b} \times \mathrm{d})+(\mathrm{a} \times \mathrm{b}) \cdot(\mathrm{c} \times \mathrm{d})$ is always equal to