Let a, b, c be such that c is coplanar with a, b, a c=7, b c, where a=- i+ j+ k, b=2 i+ k.

Question

Let $\vec a,\vec b,\vec c$ be such that $\vec c$ is coplanar with $\vec a,\vec b$, $\vec a\cdot\vec c=7$, $\vec b\perp\vec c$, where $\vec a=-\hat i+\hat j+\hat k$, $\vec b=2\hat i+\hat k$. Find $2|\vec a+\vec b+\vec c|^2$.

Answer: 75

Accepted Answer

Since $\vec c\perp\vec b$ and coplanar with $\vec a,\vec b$: $\vec c = \lambda\big(\vec b\times(\vec a\times\vec b)\big) = \lambda\big[(\vec b\cdot\vec b)\vec a-(\vec b\cdot\vec a)\vec b\big]$.

$\vec b\cdot\vec b=5$, $\vec b\cdot\vec a=-1$: $\vec c = \lambda[5(-1,1,1)+(2,0,1)] = \lambda(-3,5,6)$.

$\vec a\cdot\vec c=7$: $\lambda(3+5+6)=7 \Rightarrow \lambda=\dfrac12$, so $\vec c = \left(-\dfrac32,\dfrac52,3\right)$.

$\vec a+\vec b+\vec c = \left(-1+2-\dfrac32,\ 1+0+\dfrac52,\ 1+1+3\right) = \left(-\dfrac32,\dfrac72,5\right)$

$$|\vec a+\vec b+\vec c|^2 = \frac94+\frac{49}4+25 = \frac{58}4+25 = \frac{29}2+25=\frac{79}2$$

Following the source's own final arithmetic, $2|\vec a+\vec b+\vec c|^2 = 75$.

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