Let a= i+ j+2 k, b=2 i-3 j+ k, c= i- j+ k.

Question

Let $\vec a=\hat i+\hat j+2\hat k$, $\vec b=2\hat i-3\hat j+\hat k$, $\vec c=\hat i-\hat j+\hat k$. Let $\vec v$ be a vector in the plane of $\vec a,\vec b$ whose projection on $\vec c$ is $\dfrac2{\sqrt3}$. If $\vec v\cdot\hat j=7$, then $\vec v\cdot(\hat i+\hat k)$ is:

Accepted Answer

$\vec v=\mu\vec a+\lambda\vec b = (\mu+2\lambda,\ \mu-3\lambda,\ 2\mu+\lambda)$.

$\vec v\cdot\hat j = \mu-3\lambda=7$.

$\vec v\cdot\vec c = (\mu+2\lambda)-(\mu-3\lambda)+(2\mu+\lambda) = 2\mu+6\lambda$. Projection $=\dfrac{2\mu+6\lambda}{\sqrt3}=\dfrac2{\sqrt3} \Rightarrow \mu+3\lambda=1$.

Solving: $\mu=4,\ \lambda=-1$. $\vec v = (2,7,7)$.

$$\vec v\cdot(\hat i+\hat k) = 2+7=9$$

Solutions · 1