There are two unit vectors $\vec{a}$ and $\vec{b}$ such that $|\vec{a}+\vec{b}|=\sqrt{3}$ and $\theta$ is the angle between $\vec{a}$ and $\vec{b}$, then

Match the LIST-I with LIST-II

	LIST-I		LIST-II
A.	$\vert \vec{a} - \vec{b} \vert$	I.	$0$
B.	$\cos\theta$	II.	$1$
C.	$\theta$	III.	$\dfrac{1}{2}$
D.	$\sin 3\theta$	IV.	$\dfrac{\pi}{3}$

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Which of the following statements are correct?

A. The unit vector in the direction of the vector $\vec{a}=\hat{i}+\hat{j}+2\hat{k}$ is $\frac{1}{\sqrt{6}}\hat{i}+\frac{1}{\sqrt{6}}\hat{j}+\frac{2}{\sqrt{6}}\hat{k}$.

B. A vector in the direction of the vector $5\hat{i}-\hat{j}+2\hat{k}$ which has magnitude 8 units is $40\hat{i}-8\hat{j}+16\hat{k}$.

C. The vector joining the points P(2, 3, 0) and Q(–1, –2, –4) directed from P to Q is $-3\hat{i}-5\hat{j}-4\hat{k}$.

D. The position vector of the mid-point of the vector joining the points P(2, 3, 4) and Q(4, 1, –2) is $3\hat{i}+2\hat{j}-\hat{k}$.

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For the vector $\vec a = \dfrac{1}{3}(2\hat i - 2\hat j + \hat k)$, which of the following statements is/are TRUE ?

A. vector $\vec a$ is a unit vector

B. vector $\vec a$ is making an angle $\dfrac{\pi}{3}$ with vector $2\hat i - 4\hat j + 3\hat k$

C. vector $\vec a$ is parallel to vector $-\hat i + \hat j - \dfrac{1}{2}\hat k$

D. vector $\vec a$ is perpendicular to vector $3\hat i + 2\hat j - 2\hat k$

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If a unit vector $\hat a$ makes an angle $\dfrac{\pi}{3}$ with $\hat i$, $\dfrac{\pi}{4}$ with $\hat j$ and an acute angle $\theta$ with $\hat k$, then the respective value of $\theta$ and $\hat a$ are

If $|\vec{a}| = 1$, $|\vec{b}| = 2$, $|2\vec{a}+\vec{b}| = 2\sqrt{3}$ then $|\vec{a}-\vec{b}|$ is:

Let $\vec{a} = (\hat{i}+4\hat{j}+2\hat{k})$, $\vec{b} = (3\hat{i}-2\hat{j}+7\hat{k})$, $\vec{c} = (2\hat{i}-\hat{j}+4\hat{k})$. A vector $\vec{d}$, which is perpendicular to both $\vec{a}$ and $\vec{b}$, such that $\vec{c}.\vec{d} = 18$, is

If $\vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c}$, $\vec{a} \times \vec{b} = \vec{a} \times \vec{c}$ and $\vec{a} \neq {0}$, then the vector $\vec{b}$ in equal to.