Practicing Success
A vector coplanar with vectors $\hat i +\hat j$ and $\hat j + \hat k$ and parallel to the vector $2\hat i-2\hat j-4\hat k$, is |
$\hat i-\hat k$ $\hat i-\hat j-2\hat k$ $\hat i+\hat j-\hat k$ $3\hat i+3\hat j-6\hat k$ |
$\hat i-\hat j-2\hat k$ |
Any vector coplanar with vectors $\hat i +\hat j$ and $\hat j + \hat k$ is $\vec a=x(\hat i +\hat j)+y(\hat j + \hat k)$ or, $\vec a=x\hat i+(x+y)\hat j+y\hat k$ It is given that $\vec a$ is parallel to $2\hat i-2\hat j-4\hat k$ $∴\vec a=λ(2\hat i-2\hat j-4\hat k)$ for some scalar λ $⇒\{x\hat i+(x+y)\hat j+y\hat k\}=λ(2\hat i-2\hat j-4\hat k)$ $⇒x=2λ,x+y=-2λ$ and $y=-4λ$ $⇒x=2λ$ and $y=-4λ$ $∴\vec a=2λ(\hat i-\hat j-2\hat k)$, where $λ∈R$ Hence, option (2) is true. |