Vectors

If $\vec{a}, \vec{b}, \vec{c}$ are unit vectors, then $|\vec{a}-\vec{b}|^2 + |\vec{b}-\vec{c}|^2 + |\vec{c}-\vec{a}|^2$ does not exceed:

$ \vec{v} = 2\hat{i} + \hat{j} - \hat{k},\quad \vec{w} = \hat{i} + 3\hat{k} $ If $ \vec{u} $ is a unit vector, maximum value of $ [\vec{u}\ \vec{v}\ \vec{w}] $ is:

Let $\vec{a}, \vec{b}, \vec{c}$ be the position vectors of three vertices A, B, C of a triangle respectively then the area of this triangle is given by

For the vectors $\vec{a}=-4\hat{i}+2\hat{j}, \vec{b}=2\hat{i}+\hat{j}$ and $\vec{c}=2\hat{i}+3\hat{j}$, if $\vec{c}=m\vec{a}+n\vec{b}$ then the value of m + n is

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}, \vec{b}=\hat{i}-\hat{j}+\hat{k}$ and $\vec{c}=\hat{i}-\hat{j}-\hat{k}$ be three vectors. A vector $\vec{v}$ in the plane of $\vec{a}$ and $\vec{b}$ whose projection on $\frac{\vec{c}}{|\vec{c}|}$ is $\frac{1}{\sqrt{3}}$, is

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

Let $\vec{a}$ and $\vec{b}$ be two vectors, which of the following vectors are not perpendicular to each other?

The direction cosines of the vector a = (- 2i + j – 5k) are

A particle P starts from the point z₀=1+2i, where i=√−1 . It moves first horizontally away from origin by 5 units and then vertically away from origin by 3 units to reach a point z₁. From z₁ the particle moves √2 units in the direction of the vector $\hat{i}+\hat{j}$ and then it moves through an angle $\dfrac{\pi}{2}$ in anticlockwise direction on a circle with centre at origin, to reach a point z₂. The point z₂ is given by