Let a= i+2 j+3 k, b=3 i+ j- k, and c be a vector coplanar with a and b.

Question

Let $\vec a=\hat i+2\hat j+3\hat k$, $\vec b=3\hat i+\hat j-\hat k$, and $\vec c$ be a vector coplanar with $\vec a$ and $\vec b$. If $\vec c$ is perpendicular to $\vec b$, and $\vec a\cdot\vec c=5$, then $|\vec c|$ is equal to:

Accepted Answer

Since $\vec c$ is coplanar with $\vec a,\vec b$ and perpendicular to $\vec b$: $\vec c = \lambda\big(\vec b\times(\vec a\times\vec b)\big) = \lambda\big((\vec b\cdot\vec b)\vec a - (\vec a\cdot\vec b)\vec b\big)$.

$\vec b\cdot\vec b=11$, $\vec a\cdot\vec b = 3+2-3=2$: $\vec c = \lambda(11\vec a-2\vec b) = \lambda(5\hat i+20\hat j+35\hat k) = 5\lambda(\hat i+4\hat j+7\hat k)$.

$\vec a\cdot\vec c = 5\lambda(1+8+21)=150\lambda = 5 \Rightarrow \lambda=\dfrac1{30}$, so $\vec c = \dfrac16(\hat i+4\hat j+7\hat k)$.

$$|\vec c|^2 = \frac{1+16+49}{36} = \frac{66}{36}=\frac{11}6 \implies |\vec c|=\sqrt{\frac{11}6}$$

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