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

Which of the following statements are correct ?

A. If $\vec{a}$ and $\vec{b}$ are two adjacent sides of a triangle, then the area of the triangle is $\frac{1}{2}|\vec{a}\times\vec{b}|$

B. If $\vec{a}$ and $\vec{b}$ are two adjacent sides of a triangle, then the area of the triangle is $\frac{1}{2}|\vec{a}.\vec{b}|$

C. If $\vec{a}$ and $\vec{b}$ are two adjacent sides of a parallelogram, then its area is $|\vec{a}\times\vec{b}|$

D. If $\vec{a}$ and $\vec{b}$ are two adjacent sides of a parallelogram, then its area is $|\vec{a}.\vec{b}|$

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If $\vec{a}$ is a non-zero vector, then always

If $\hat{a}, \hat{b}$ and $\hat{c}$ are three unit vectors and $\hat{a} + \hat{b} + \hat{c} = \vec{0}$, then the angle between $\hat{a}$ and $(-\hat{b})$ is

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:

If $\vec{a} = 2\hat{i} - 2\hat{j} + \hat{k}$, then which of the following statements are TRUE?

(A) unit vector along $\vec{a}$ is $\frac{1}{3}(2\hat{i}+2\hat{j}-\hat{k})$

(B) $|3\vec{a}| = 9$

(C) $\vec{a}$ is perpendicular to $2\hat{i}+3\hat{j}+2\hat{k}$

(D) Direction Ratios of $\vec{a}$ are 2,-2,1

Choose the correct answer from the options given below: