If $\vec{a}=-2 \hat{i}+9 \hat{j}-6 \hat{k}$ and $\vec{b}=t \hat{i}-2 \hat{j}+6 \hat{k}$ are vectors such…
If $\vec{a}=-2 \hat{i}+9 \hat{j}-6 \hat{k}$ and $\vec{b}=t \hat{i}-2 \hat{j}+6 \hat{k}$ are vectors such that $|\vec{a}+\vec{b}|=25$, then the sum of the values of $t$ is
$14$
$11$
$4$
$77$
Solution
Given, $\vec{a}=-2 \hat{i}+9 \hat{j}-6 \hat{k}, \hat{b}=t \hat{i}-2 \hat{j}+6 \hat{k}$
$\begin{aligned}
& \text { Now, }|\vec{a}+\vec{b}|=25 \\
& \Rightarrow|(t-2) \hat{i}+7 \hat{j}+0 \hat{k}|=25 \Rightarrow \sqrt{(t-2)^2+49+0^2}=25 \\
& \Rightarrow(t-2)^2+49=625 \Rightarrow(t-2)^2=576 \\
& \Rightarrow t^2-4 t+4=576
\end{aligned}$
So sum of value of $t=4$