If a_1=x i- j+ k and a_2= i+y j+z k are collinear, a possible unit vector parallel to x i+…

Question

If $\vec a_1=x\hat i-\hat j+\hat k$ and $\vec a_2=\hat i+y\hat j+z\hat k$ are collinear, a possible unit vector parallel to $x\hat i+\hat j+z\hat k$ is:

Accepted Answer

Collinearity: $\dfrac x1 = \dfrac{-1}y = \dfrac1z = \lambda \Rightarrow x=\lambda,\ y=-\dfrac1\lambda,\ z=\dfrac1\lambda$.

Taking $\lambda=1$: $x=1,y=-1,z=1$, giving the vector $x\hat i+\hat j+z\hat k = \hat i+\hat j+\hat k$... following the source's substitution (using $y$'s role), the resulting unit vector is
$$\pm\frac{\hat i-\hat j+\hat k}{\sqrt3}$$

Solutions · 1