JEE Main 2005 — Vectors & 3D Geometry

Let $\vec{a}=\hat{i}-\hat{k}, \vec{b}=x \hat{i}+\hat{j}+(1-x) \hat{k}$ and $\vec{c}=y \hat{i}+x \hat{j}+(1+x-y) \hat{k}$. Then $[\vec{a}, \vec{b}, \vec{c}]$ depends on

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and $\lambda$ is a real number then $\left[\lambda(\vec{a}+\vec{b}) \lambda^2 \vec{b} \lambda \vec{c}\right]=[\vec{a} \vec{b}+\vec{c} \vec{b}]$ for

For any vectora a the value of $(\vec{a} \times \hat{i})^2+(\vec{a} \times \hat{j})^2+(\vec{a} \times \hat{k})^2$ is equal to

Let $a, b$ and $c$ be distinct non-negative numbers. If the vectors $a \hat{i}+a \hat{j}+c \hat{k}, \hat{i}+\hat{k}$ and $c \hat{i}+c \hat{j}+b \hat{k}$ lie in a plane, then $c$ is

The resultant $R$ of two forces acting on a particle is at right angles to one of them and its magnitude is one third of the other force. The ratio of larger force to smaller one is

If $C$ is the mid point of $A B$ and $P$ is any point outside $A B$, then

The line parallel to the $x$-axis and passing through the intersection of the lines ax $+$ $2 b y+3 b=0$ and $b x-2 a y-3 a=0$, where $(a, b) \neq(0,0)$ is