If $a$ and $b$ are vectors such that $|a+b| = \sqrt{29}$ and $a \times (2i+3j+4k) = (2i+3j+4k) \times b$, then a possible value of $(a+b) \cdot (-7i+2j+3k)$ is

If a⃗ and b⃗ are vectors such that |a⃗+b⃗|=√29 and a⃗×(2î+3ĵ+4k̂)=(2î+3ĵ+4k̂)×b⃗, then a possible value of (a⃗+b⃗)·(-7î+2ĵ+3k̂) is

$\left|\vec a+\vec b\right| = \sqrt{29}$

$\vec a\times\vec c = \vec c\times\vec b,\quad \vec c = 2\hat i+3\hat j+4\hat k$

$\text{Find a possible value of } (\vec a+\vec b)\cdot(-7\hat i+2\hat j+3\hat k)$

$1.\ (\vec a+\vec b)\times\vec c = \vec 0 \Rightarrow (\vec a+\vec b)\parallel\vec c$

$2.\ |\vec a+\vec b|=|\vec c| \Rightarrow \vec a+\vec b=\pm\vec c$

$3.\ \text{evaluate } \vec c\cdot(-7\hat i+2\hat j+3\hat k)$

$\vec a\times\vec c = \vec c\times\vec b = -\vec b\times\vec c$

$\Rightarrow\ \vec a\times\vec c + \vec b\times\vec c = (\vec a+\vec b)\times\vec c = \vec 0$

$\vec a+\vec b \parallel \vec c$

$|\vec c| = \sqrt{4+9+16} = \sqrt{29} = |\vec a+\vec b| \Rightarrow \vec a+\vec b = \pm\vec c$

$\vec c\cdot\vec w = 2(-7)+3(2)+4(3) = -14+6+12 = 4$

$(\vec a+\vec b)\cdot\vec w = \pm(\vec c\cdot\vec w) = \pm 4$

$\boxed{4 \Rightarrow \text{option (3)}}$