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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If $|\vec{a}| = \sqrt{26}, |\vec{b}| = 7, |\vec{a}\times\vec{b}| = 35$ and $\theta$ be the angle between $\vec{a}$ and $\vec{b}$ then

Match the LIST-I with LIST-II

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If $\vec a\times\vec b=\vec a\times\vec c$ and $\vec a\neq\vec 0$, then which of the following is/are TRUE?

A. $\vec b=\vec c$

B. $\vec b+\vec c=\vec 0$

C. $\vec b=\vec c+\lambda\vec a$ for some scalar λ

D. $\vec b+\vec c=\lambda\vec a$ for some scalar λ

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The angle between the vectors $\hat{i} - \hat{j}$ and $\hat{j} - \hat{k}$ is:

If $\theta$ is the acute angle between two unit vectors $\vec{a}$ and $\vec{b}$, then $\cos\frac{\theta}{2} =$

If $\vec{p}$ and $\vec{q}$ are two unit vectors such that $|\vec{p} + \vec{q}| = \sqrt{2}$, then which of the following are correct?

(A) $|\vec{p}| = |\vec{q}| = 1$

(B) $\vec{p}$ and $\vec{q}$ are orthogonal vectors

(C) $\vec{p}$ and $\vec{q}$ are collinear vectors

(D) $(4\vec{p} - \vec{q}).(2\vec{p} + \vec{q}) = 7$

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Which of the following statements are true?

(A) If $\vec{r} = x\hat{i} + y\hat{j} + z\hat{k}$, then $x, y, z$ are called direction ratios of $\vec{r}$.

(B) For any two vectors $\vec{a}$ and $\vec{b}$, $\vec{a} + \vec{b} = \vec{b} + \vec{a}$

(C) $\vec{a} \perp \vec{b}$ if and only if $\vec{a} \times \vec{b} = \vec{0}$

(D) Projection of $\vec{b}$ on $\vec{a}$ is $\frac{\vec{a} \cdot \vec{b}}{|\vec{a}|^2}$

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