The length of line segment joining the points with position vectors $2\hat{i} - 2\hat{j} + 3\hat{k}$ and $5\hat{i} + 2\hat{j} + 3\hat{k}$ is

Match List-I with List-II

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If $\overrightarrow{AB}=2\hat i+3\hat j-\hat k$ and $\overrightarrow{AC}=3\hat i-2\hat j+4\hat k$ are the two sides of a triangle ABC, then the length of the median through vertex A is equal to

Let $\vec{a} = 3\hat{i} + \hat{j} - 4\hat{k}$ and $\vec{b} = 6\hat{i} + 5\hat{j} - 2\hat{k}$ be two vectors. Then a vector perpendicular to $\vec{a}$ and $\vec{b}$ with magnitude 3 units is

Let $\vec{a} = \hat{i} + 4\hat{j}$, $\vec{b} = 4\hat{j} + \hat{k}$ and $\vec{c} = \hat{i} - 2\hat{k}$. If $\vec{d}$ is a vector perpendicular to both $\vec{a}$ and $\vec{b}$ such that $\vec{c} \cdot \vec{d} = 16$, then $|\vec{d}|$ is equal to

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 $\cos^{-1}\left(\dfrac{5}{7}\right)$

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If $\vec{a}$ and $\vec{b}$ are two perpendicular vectors such that $|\vec{a}| = 3$, $|\vec{b}| = 4$ and $\theta$ is the angle between $\vec{a}$ and $(\vec{a} - \vec{b})$, then $\cos\theta$ is equal to: