Vectors Questions

The vectors $\overrightarrow{AB}=3\hat{i}+4\hat{k}$ and $\overrightarrow{AC}=5\hat{i}-2\hat{j}+4\hat{k}$ are the sides of a triangle $ABC$. The length of the median through $A$ is:

If $\vec{a}=2\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=-\hat{i}+2\hat{j}+\hat{k}$ and $\vec{c}=3\hat{i}+\hat{j}$ are such that $\vec{a}+\gamma \vec{b}$ is perpendicular to $\vec{c}$, then determine the value of $\gamma$?

A unit vector $\vec{a}$ makes angle $\pi /3$ with $\hat{i}$, $\pi /4$ with $\hat{j}$ and an acute angle $\theta$ with $\hat{k}$.I. The value of $\theta$ is $\pi /3$.II. The components of $\vec{a}$ are $(1/2,1/2,1/2)$.Which of the above statement(s) is/are correct?

If the vertices $A,B,C$ of a triangle $ABC$ are $(1,1,3)$, $(-1,0,0)$, $(0,1,2)$ respectively, then determine $\angle ABC$.

In a triangle $ABC$, $|\overrightarrow{BC}|=8$, $|\overrightarrow{CA}|=7$, $|\overrightarrow{AB}|=10$, then the projection of the $\overrightarrow{AB}$ on $\overrightarrow{AC}$ is equal to:

If $\vec{a}=4\hat{i}+6\hat{j}$ and $\vec{b}=3\hat{j}+4\hat{k}$ then the component of $\vec{a}$ along $\vec{b}$ is:

The vectors $\vec{a}=x\hat{i}+y\hat{j}+z\hat{k}$ and $\vec{b}=\hat{j}$ are such that $\vec{a},\vec{c},\vec{b}$ form a right-handed system. Then $\vec{c}$ is:

Let $a,b,c$ be distinct non-negative numbers. If the vectors $a\hat{i}+a\hat{j}+c\hat{k}$, $\hat{i}+\hat{k}$ and $c\hat{i}+c\hat{j}+b\hat{k}$ lie in a plane, then $c$ is: