Vector Algebra

The position vectors of two adjacent sides $\overrightarrow{O A}$ and $\overrightarrow{O B}$ of a rectangle $O A C B$ are $\vec{a}$ and $\vec{b}$ respectively, where $O$ is the origin. If $16|\vec{a} \times \vec{b}|=3(|\vec{a}|+|\vec{b}|)^2...

If $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{i}-\hat{j}+\hat{k}, \vec{c}=\hat{i}+2 \hat{j}-\hat{k}$, then the value of $\left|\begin{array}{lll}\vec{a} \cdot \vec{a} & \vec{a} \cdot \vec{b} & \vec{a} \cdot \vec{c} \\ \vec{b} \cdot \vec...

Let $\vec{r}=\sin x(\vec{a} \times \vec{b})+\cos y(\vec{b} \times \vec{c})+2(\vec{c} \times \vec{a})$ ，where $\vec{a}, \vec{b}$ and $\vec{c}$ are three non－coplanar vectors． It is given that $\vec{r}$ is perpendicular to $(\vec{a}+\vec{b}+\...

Let $\vec{a}=(x, y, z)$ be the vector with $|\vec{a}|=2 \sqrt{3}$, which makes equal angles with the vector $\vec{b}=(y,-2 z, 3 x)$ and $\vec{c}=(2 z, 3 x,-y)$ and is perpendicular to the vector $\vec{d}=(1,-1,2)$. If the angle between $\ve...

A vector given by $\vec{P}=f(t) \hat{i}+g(t)+\hat{k}$ moves in such a way that it is always parallel to the vector $\vec{Q}=-f^{\prime \prime}(t) \hat{i}+f^{\prime}(t) \hat{j}+\hat{k}$.

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

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

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 \a...

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

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 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 volum...

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 ...

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 \over...

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 $$\overright...

If a(\(\alpha\) \(\times\) \(\beta\)) + b(\(\beta\) \(\times\) \(\gamma\)) + c(\(\gamma\) + \(\alpha\)) = 0, where a, b, c are non-zero scalars, then the vectors \(\alpha\), \(\beta\), \(\gamma\) are

let \(\alpha\), \(\beta\), \(\gamma\) be three non-zero vectors which are pairwise non-collinear. if \(\alpha\) + 3\(\beta\) is collinear with \(\gamma\) and \(\beta\) + 2\(\gamma\) is collinear with \(\alpha\) then \(\alpha\) + 3\(\beta\) ...

The unit vector in ZOX plane, making angles \(45^\circ\) and \(60^\circ\) respectively with \(\alpha = 2\widehat i + 2\widehat j - \widehat k\) and \(\beta = \widehat j - \widehat k\) is

The position vectors of the points A, B, C and D are \(3\widehat i - 2\widehat j - \widehat k\), \(2\widehat i - 3\widehat j + 2\widehat k\), \(5\widehat i - \widehat j + 2\widehat k\) and \(4\widehat i - \widehat j - \lambda \widehat k\), ...

Let \(\widehat \alpha\), \(\widehat \beta\), \(\widehat \gamma\) be three unit vectors such that \(\widehat \alpha \, \times \,(\widehat \beta \times \widehat \gamma ) = {1 \over 2}(\widehat \beta + \widehat \gamma )\) where $$\widehat...

Let \(\overrightarrow \alpha\) = \(\widehat i + \widehat j + \widehat k\), \(\overrightarrow \beta\) = \(\widehat i - \widehat j - \widehat k\) and \({\overrightarrow \gamma }\) = \(- \widehat i - \widehat j - \widehat k\) be three ve...

Let \(\overrightarrow \alpha\), \({\overrightarrow \beta }\), \({\overrightarrow \gamma }\) be the three unit vectors such that \(\overrightarrow \alpha .\overrightarrow \beta = \overrightarrow \alpha .\overrightarrow \gamma = 0\)...

If the sum of two unit vectors is a unit vector, then the magnitude of their difference is

For any vector x, where \(\widehat i\), \(\widehat j\), \(\widehat k\) have their usual meanings the value of \({(x \times \widehat i)^2} + {(x \times \widehat j)^2} + {(x \times \widehat k)^2}\) where \(\widehat i\), \(\widehat j\), $$\wid...