The area (in sq.units) of a triangle formed by vertices O, A and B where $\vec{OA} = \hat{i} + 2\hat{j} + 3\hat{k}$ and $\vec{OB} = -3\hat{i} - 2\hat{j} + \hat{k}$ is

Which of the following statements are correct?

(A) $\hat{i} \times (\hat{j} \times \hat{k}) = \vec{0}$.

(B) If $|\vec{a}\times\vec{b}|=2$, $\vec{a}.\vec{b}=4$, then $|\vec{a}|^2|\vec{b}|^2=6$.

(C) The number of vectors of unit length perpendicular to $\vec{a}$ and $\vec{b}$ are 2.

(D) If $\vec{a}$ and $\vec{b}$ represent the diagonals of the rhombus, then $\vec{a}.\vec{b}=0$

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Which of the following statement are true?

A. The angle between the vectors $2\hat i+\hat j+3\hat k$ and $3\hat i-2\hat k$ is $\frac{\pi}{2}$

B. If $\vec a$ and $\vec b$ are two non-zero vectors, then projection of $\vec b$ on $\vec a$ is $\frac{\vec a\cdot\vec b}{|\vec a|}$

C. The unit vector normal to both vectors $\vec a=\hat i-\hat j-\hat k$ and $\vec b=\hat i+\hat j+\hat k$ is $\frac{1}{\sqrt2}(-\hat j+\hat k)$

D. The area of triangle formed by adjacent sides represented by vectors $\vec a=3\hat i+4\hat j$ and $\vec b=-5\hat i+7\hat j$ is 41 sq. units

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A unit vector perpendicular to both the vectors $4\hat{i}-\hat{j}+3\hat{k}$ and $-2\hat{i}+\hat{j}-2\hat{k}$ is:

If $|\vec{a}| = 3|\vec{b}|$, $|\vec{b}| = 2$ and angle between $\vec{a}$ and $\vec{b}$ is $60^\circ$, then $|\vec{a} - \vec{b}|$ is equal to:

Which of the following statements are correct?

(A) If $\vec{a}$ and $\vec{b}$ represent the adjacent sides of a triangle, then its area is $\frac{1}{2}|\vec{a} \times \vec{b}|$

(B) If $\vec{a}$ and $\vec{b}$ represent the adjacent sides of a parallelogram, then its area is $|\vec{a} \times \vec{b}|$

(C) $|\vec{a} \times \vec{b}| = |\vec{a}| |\vec{b}| \cos\theta$

(D) If $\vec{a}$ and $\vec{b}$ represent the 'diagonals' of a parallelogram, then its area is $\frac{1}{2}|\vec{a} \times \vec{b}|$

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If the vectors $\vec{a} = 3\hat{i} - p\hat{j} + 5\hat{k}$ and $\vec{b} = -6\hat{i} + 14\hat{j} + q\hat{k}$ are collinear, then the value of p and q are: