If $a = i + 2j - 3k$ and $b = 3i - j + 2k$,find the angle between the vectors $a + b$ and $a - b$ in degrees.

The angle between the diagonals of the parallelogram whose adjacent sides are $2 \hat{i}+4 \hat{j}-5 \hat{k}$ and $\hat{i}+2 \hat{j}+3 \hat{k}$ is

Let $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$ be the position vector of a point $A$. Let $\vec{b}=\hat{i}+2 \hat{j}-\hat{k}$ and $\vec{c}=\hat{i}+\hat{j}-2 \hat{k}$ be two vectors and $\vec{r}$ be a vector passing through the point $A$ with position vector $\vec{a}$ and parallel to the vector $\vec{b}$. If the projection of $\vec{r}$ on $\vec{c}$ is $\frac{9}{\sqrt{6}}$,then find $|\vec{r}|$.

If $A(3,4,5), B(4,6,3), C(-1,2,4)$ and $D(1,0,5)$ are such that the angle between the lines $DC$ and $AB$ is $\theta$,then $\cos \theta$ is equal to

If $a, b, c$ are unit vectors such that $a + b + c = 0,$ then $a \cdot b + b \cdot c + c \cdot a = $

Let $a, b, c$ be three vectors such that the magnitude of $b$ is twice that of $a$ and the magnitude of $c$ is three times that of $a$. If the angle between each pair of vectors is $\frac{\pi}{3}$ and $|a+b+c|=5$,then $|c|+|a|+|b|=$