If a=2 i+ j+3 k, b=3 i+3 j+ k, c=c_1 i+c_2 j+c_3 k are coplanar, a c=5, and b c, then 122(…

Question

If $\vec a=2\hat i+\hat j+3\hat k$, $\vec b=3\hat i+3\hat j+\hat k$, $\vec c=c_1\hat i+c_2\hat j+c_3\hat k$ are coplanar, $\vec a\cdot\vec c=5$, and $\vec b\perp\vec c$, then $122(c_1+c_2+c_3)$ is:

Answer: 150

Accepted Answer

$\vec b\cdot\vec c=0$: $3c_1+3c_2+c_3=0$ \ldots  (A)

$\vec a\cdot\vec c=5$: $2c_1+c_2+3c_3=5$ \ldots  (B)

Coplanar with $\vec a,\vec b$: $\vec c$ lies in the plane through origin normal to $\vec a	imes\vec b=(-8,7,3)$: $-8c_1+7c_2+3c_3=0$ \ldots  (C)

Solving (A),(B),(C): $c_1=\dfrac{10}{122},\ c_2=-\dfrac{85}{122},\ c_3=\dfrac{225}{122}$.

$122(c_1+c_2+c_3) = 10-85+225 = 150$

Solutions · 1