Let $\vec{a}=\hat{i}-\alpha \hat{j}+\beta \hat{k}, \vec{b}=3 \hat{i}+\beta \hat{j}+\alpha \hat{k}$ and $\vec{c}=-\alpha \hat{i}+2 \hat{j}+\hat{k}$, where $\alpha$ and $\beta$ are integers. If $\vec{a} \cdot \vec{b}=-1$ and $...

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a}=\vec{b} \times(\vec{b} \times \vec{c})$. If magnitudes of the vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are $...

For real numbers $\alpha$ and $\beta \neq 0$, if the point of intersection of the straight lines $\frac{x-\alpha}{1}=\frac{y-1}{2}=\frac{z-1}{3}$ and $\frac{x-4}{\beta}=\frac{y-6}{3}=\frac{z-7}{3}$, lies on the plane $...

If a point R(4, y, z) lies on the line segment joining the points P(2, –3, 4) and Q(8, 0, 10), then the distance of...

Let the vectors $(2+a+b) \hat{i}+(a+2 b+c) \hat{j}-(b+c) \hat{k},(1+b) \hat{i}+2 b \hat{j}-b \hat{k}$ and $(2+b) \hat{i}+2 b \hat{j}+(1-b) \hat{k}, a, b, c \in R$ be co-planar....

If the lines $\frac{x-k}{1}=\frac{y-2}{2}=\frac{z-3}{3}$ and $\frac{x+1}{3}=\frac{y+2}{2}=\frac{z+3}{1}$ are co-planar, then the value of $k$ is

If $(\vec{a}+3 \vec{b})$ is perpendicular to $(7 \vec{a}-5 \vec{b})$ and $(\vec{a}-4 \vec{b})$ is perpendicular to $(7 \vec{a}-2 \vec{b})$, then the angle between $\vec{a}$ and $\vec{b}$...

The vector equation of plane through the line of intersection of the planes $x+y+z=1$ and $2 x+3 y+4 z=5$ which is perpendicular to the plane...

Let a plane $P$ pass through the point $(3,7,-7)$ and contain the line, $\frac{x-2}{-3}=\frac{y-3}{2}=\frac{z+2}{1}$. If distance of the plane P from the origin is d...