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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:

Vector projection of $7\hat i+\hat j+4\hat k$ on the vector $2\hat i+6\hat j-3\hat k$ is

If $\vec a$ and $\vec b$ are two unit vectors and $|\vec a-\vec b|=\sqrt3$, then the value of $|\vec a+\vec b|$ is:

If $\vec a, \vec b$ and $\vec c$ are three vectors, then meaningless expressions are

(A) $\vec a.(\vec b\times\vec c)$

(B) $\vec a.(\vec b.\vec c)$

(C) $\vec a\times(\vec b\times\vec c)$

(D) $\vec a\times(\vec b.\vec c)$

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If $|\vec a|=2, |\vec b|=7$ and $\vec a\times\vec b=3\hat i+2\hat j+6\hat k$, then angle between $\vec a$ and $\vec b$ is:

If $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $|\vec{a}| = 2, |\vec{b}| = 5, |\vec{c}| = 3$ and $\vec{a} + 2\vec{b} - 3\vec{c} = \vec{0}$, then angle between $\vec{b}$ and $\vec{c}$ is equal to:

Let $\vec{a}$ and $\vec{b}$ be any two vectors such that $\vec{a} = 2\hat{i} + \hat{j} + 3\hat{k}$ and $\vec{b} = 3\hat{i} + 5\hat{j} - 2\hat{k}$ then

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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 $\vec{a}, \vec{b}, \vec{c}$ are unit vectors such that $\vec{a} + \vec{b} + \vec{c} = 0$. The value of $\vec{a}.\vec{b} + \vec{b}.\vec{c} + \vec{c}.\vec{a}$ is

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

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Let $\vec{a}$ be any vector such that $\vec{a}.\hat{i} = \vec{a}.(\hat{i}+\hat{j}) = \vec{a}.(\hat{i}+\hat{j}+\hat{k}) = 1$, then $\vec{a}$ is

Let us consider vector $\vec{a} = (c\log_2 x)\hat{i} - 6\hat{j} + 3\hat{k}$ and $\vec{b} = (\log_2 x)\hat{i} + 2\hat{j} + (2c\log_2 x)\hat{k}$, then which of following statements are TRUE?

(Where $x \in \mathbb{R}$ (set of real numbers))

(A) Vector $\vec{a}$ and $\vec{b}$ make obtuse angle for any $x \in (0,\infty)$ if $c \in (-1,0)$

(B) Vector $\vec{a}$ and $\vec{b}$ makes obtuse angle for any $x \in \mathbb{R}$ if $c > 0$

(C) Vector $\vec{a}$ and $\vec{b}$ makes obtuse angle for any $x \in (0,\infty)$ if $c \in \left(-\frac{4}{3}, -1\right]$

(D) Vectors $\vec{a}$ and $\vec{b}$ makes obtuse angle for $x \in \mathbb{R}$ if $c \in (1,2)$

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Let $\vec{a}$ and $\vec{b}$ be two vectors, such that $|\vec{a}| = 3, |\vec{b}| = 5$ and the angle between them is $60°$, then the value of $|\vec{a} + \vec{b}|$ is

If $\theta$ is angle between $\vec{a}$ & $\vec{b}$, then for vectors $\vec{a}, \vec{b}$ and $\vec{c}$,

Match the LIST-I with LIST-II

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Vectors $\vec{a}, \vec{b}, \vec{c}$ are such that $\vec{a} + \vec{b} + \vec{c} = 0$ and $|\vec{a}| = |\vec{b}| = |\vec{c}|$ then, the angle between any two of them, is

If $\vec{a}$ and $\vec{b}$ are two collinear vectors, then which of the following statements is/are incorrect?

(A) $\vec{b} = \lambda\vec{a}$, for some scalar $\lambda$.

(B) Always $\vec{a} = \pm\vec{b}$.

(C) the respective components of $\vec{a}$ and $\vec{b}$ are proportional.

(D) both vectors $\vec{a}$ and $\vec{b}$ always have same direction.

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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

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The position vector of a point R which divides the line joining the points A(-2, 1, 3) and B(3, 5, -2) internally in the ratio 2:1, is

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

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The vector $\{(a^2+a-2)\hat{i}+(a^2-4)\hat{j}-(a^2-5a+6)\hat{k}\}$ makes acute angles with coordinate axes, then values of $a$ lies in :

If $P(\vec{r})$ is a position vector of a point with scalar components $(x,y,z)$, then the value of $(\vec{r}.\hat{i})(-\hat{i})-(\vec{r}.\hat{j})\hat{j}+(\vec{r}.(-\hat{k}))\hat{k}$, is:

If $\vec{a}$ and $\vec{b}$ are two non-zero vectors such that $|2\vec{a}+\vec{b}|=|\vec{b}|$ and $\vec{a}+\vec{b}\ne \vec{0}$, then the angle between $\vec{a}+\vec{b}$ and $\vec{a}$, is

If $|\vec{a}| = 2$, $|\vec{b}| = 3$ and $|2\vec{a} - \vec{b}| = 5$, then $|2\vec{a} + \vec{b}|$ equals to

If $|\vec{a}| = 5$, $|\vec{b}| = 6$ and $\vec{a}.\vec{b} = 18$, then value of $|\vec{a} \times \vec{b}|$ is

The position vector of the point which divides the join of the points $2\vec{a}-3\vec{b}$ and $3\vec{a}-2\vec{b}$ externally in the ratio 2:3, 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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If $\vec a, \vec b$ and $\vec c$ are vectors such that $|\vec a|=5, |\vec b|=12, |\vec c|=13$ and $\vec a+\vec b+\vec c=\vec 0$, then

Match List-I with List-II

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If $|\vec a|=2, |\vec b|=7, \vec a \times \vec b = 3\hat i + 2\hat j + 6\hat k$, then the angle between $\vec a$ and $\vec b$ is equal to

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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