For the vectors $\vec a=\hat i-2\hat j+3\hat k$ and $\vec b=3\hat i-2\hat j+\hat k$, Which of the following are correct?

(A) $|\vec a|=14$

(B) $|\vec a|=|\vec b|$

(C) $\vec a . \vec b=10$

(D) angle between $\vec a$ and $\vec b$ is $\cos^{-1}\left(\dfrac{5}{7}\right)$

Choose the correct answer from the options given below:

If $\vec{a}, \vec{b}, \vec{c}$ are three mutually perpendicular unit vectors, then $|\vec{a} + \vec{b} + \vec{c}|$ is equal to

If $\vec{a}, \vec{b}$ and $\vec{c}$ be vectors such that $\vec{a} + \vec{b} + \vec{c} = \vec{0}$, $|\vec{a}| = 3$, $|\vec{b}| = 5$ and $|\vec{c}| = 7$, then the angle between $\vec{a}$ and $\vec{b}$ is

Let $|\vec{a}| = 5, |\vec{b}| = 2$ and $\vec{a}\cdot\vec{b} = 6$, then the value of $|\vec{a} \times \vec{b}|$ is equal to

Match List-I with List-II

Consider two vectors $\vec{a} = \hat{i} + 2\hat{j} - \hat{k}$ and $\vec{b} = -3\hat{i} - 6\hat{j} + 3\hat{k}$, then

Choose the correct answer from the options given below:

If $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$ and $\vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k}$ are two non zero vectors inclined at an angle $\theta$, then identify the correct option out of the given options.

(a) $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}| \cdot |\vec{b}|}$

(b) $\vec{a}$ and $\vec{b}$ are perpendicular, if $a_1 b_1 + a_2 b_2 + a_3 b_3 = 0$

(c) $\vec{a}$ and $\vec{b}$ are perpendicular, if $\frac{a_1}{b_1} = \frac{a_2}{b_2} \neq \frac{c_1}{c_2}$

(d) for $\theta = \pi$, $\vec{a} \times \vec{b} = 0$

(e) $\cos\theta = \frac{|\vec{a} \times \vec{b}|}{|\vec{a}| \cdot |\vec{b}|}$

Choose the most appropriate answer from the options given below

The vector in the direction of the vector $2\hat{i} - \hat{j} - 2\hat{k}$ that has magnitude 9 units is: