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

Let $$\vec a = 2\hat i - \hat j + \hat k,\vec b = \hat i + 2\hat j - \hat k$$ and $$\overrightarrow c = \widehat i + \widehat j - 2\widehat k - 2\widehat k$$ be three vectors. A vector in the plane of $${\overrightarrow b }$$ and $${\overrightarrow c }$$, whose projection on $${\overrightarrow a }$$ is of magnitude $$\sqrt {2/3,} $$ is :

In a triangle $$ABC, D$$ and $$E$$ are points on $$BC$$ and $$AC$$ respectively, such that $$BD=2DC$$ and $$AE=3EC.$$ Let $$P$$ be the point of intersection of $$AD$$ and $$BE.$$ Find $$BP/PE$$ using vector methods.

A particle of mass m moves on the x-axis as follows: it starts from rest at t = 0 from the point x = 0, and come to rest at t = 1 at the point x = 1. NO other information is available about its motion at intermediate times ( 0 < t < 1 ). If $$\alpha $$ denotes the instantaneous acceleration of the particle, then: