Vector Algebra ...

If $$a, b, c$$ are vectors such that $$a+b+c = 0$$ and $$|a| = 7, |b| = 5, |c| = 3$$, then the angle between $$c$$ and $$b$$ is

$$(\vec r. \hat i)(\vec r \times \hat i)+ (\vec r. \hat j)(\vec r \times \hat j) +(\vec r. \hat k)(\vec r \times \hat k)$$ is equal to

If $$(\vec a\times \vec b)^2 +(\vec a. \vec b)^2 =144$$ and $$|\vec a|=4$$, then $$|\vec b|=$$

$$p\hat{i}+3\hat{j}+4\hat{k}$$ and $$\sqrt{q}\hat{i}+4\hat{k}$$ are two vectors, where $$p,q>0$$ are two scalars, then the length of the vectors is equal to

Given that $$\vec a, \vec b, \vec p, \vec q$$ are four vectors such that $$\vec a + \vec b = \mu \vec p, \vec b \cdot \vec q = 0$$ and $$(\vec b)^2 = 1,$$ where $$\mu$$ is scalar. Then $$\mid (\vec a \cdot \vec q) \vec p - (\vec p \cdot \vec q)\vec a \mid$$ is equal to

Vector $$ \vec{x} $$ is

Vector $$ \vec{z} $$ is

Vector $$ \vec{y} $$ is

If $$\vec r$$ and $$\vec s$$ are non-zero constant vectors and the scalar $$b$$ is chosen such that $$\mid \vec r + b \vec s \mid$$ is minimum, then the value of $$\mid b \vec s \mid^2 + \mid \vec r + b \vec s \mid^2$$ is equal to

$$ (\vec{P} \times \vec{B}) \times \vec{B} $$ is equal to