The area (in sq. units) of the parallelogram whose diagonals are represented by the vectors $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=3\hat{i}+4\hat{j}-\hat{k}$, is

Match List-I with List-II

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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:

Let $\vec{a} = 4\hat{i} - \hat{j} + 3\hat{k}$ and $\vec{b} = -2\hat{i} + \hat{j} - 2\hat{k}$. Then

(A) $\vec{a}$ is a unit vector

(B) $\vec{a} \times \vec{b} = -\hat{i} + 2\hat{j} + 2\hat{k}$

(C) $\vec{a}$ and $\vec{b}$ are parallel vectors

(D) $\vec{a}$ and $\vec{b}$ are neither parallel nor perpendicular vectors

Choose the correct answer from the options given below :

Let $\vec{a} = 2\hat{i} + 3\hat{j} + \hat{k}$ and $\vec{b} = 2\hat{i} - \hat{j} + 2\hat{k}$ be two vectors. Then which of the following are TRUE?

A. $\vec{a}$ and $\vec{b}$ are perpendicular

B. Projection of $\vec{a}$ on $\vec{b}$ is 1

C. $|\vec{a} \times \vec{b}| = 3\sqrt{13}$

D. $\vec{a} - \vec{b}$ is a unit vector

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If $\vec a=\hat i+\hat j+2\hat k$ and $\vec b=2\hat i+\hat j+2\hat k$, then the unit vector in the direction of $2\vec a-\vec b$ is equal to :

If $|\vec{a}| = 10$, $|\vec{b}| = 2$ and $\vec{a} \cdot \vec{b} = 12$, then value of $|\vec{a} \times \vec{b}|$ is :