If the vectors $a \hat{\mathbf{i}}+\mathbf{j}+3 \hat{\mathbf{k}}, 4 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $4 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}$ are coplanar, then $a$ is equal to

The values of $x$ for which the angle between the vectors $x^2 \hat{\mathbf{i}}+2 x \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+x \hat{\mathbf{k}}$ is obtuse lie in the interval

In $\triangle P Q R,(4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}),(2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}})$ and $(3 \hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \mathbf{k})$are$\mathbf{}$ the position vectors of the vectices $P, Q$ and $R$ respectively then, the position vector fo the point ol intersection of the angle bisector of $P$ and $Q R$ is

If $\theta$ is the angle between $\hat{\mathbf{f}}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $\hat{\mathbf{g}}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+a \hat{\mathbf{k}}$ and $\sin \theta=\sqrt{\frac{24}{28}}$, then $7 a^2+24 a=$

$\mathbf{a}=\alpha \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \quad \mathbf{b}=\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ and $\mathbf{c}=3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ ar linearly dependent vectors and magnitude of $ \alpha $$ \sqrt{14} $${\text {}}{ }^{}$ If $\alpha, \beta$ are integers, then $\alpha+\beta=$

a, b, c are non-coplanar vectors. If $$\mathbf{a}+3 \mathbf{b}+4 \mathbf{c}=x(\mathbf{a}-2 \mathbf{b}+3 \mathbf{c})+y(\mathbf{a}+5 \mathbf{b}-2 \mathbf{c}) +z(6 \mathbf{a}+14 \mathbf{b}+4 \mathbf{c}) \text {, then } x+y+z=$$

Three vectors of magnitudes $$a, 2 a, 3 a$$ are along the directions of the diagonals of 3 adjacent faces of a cube that meet in a point. Then, the magnitude of the sum of those diagonals is

If $$\mathbf{a}$$ is collinear with $$\mathbf{b}=3 \hat{i}+6 \hat{j}+6 \hat{k}$$ and $$\mathbf{a} \cdot \mathbf{b}=27$$, then $$|\mathbf{a}|=$$

Let $$a, b$$ and $$c$$ be unit vectors such that $$a$$ is perpendicular to the plane containing $$\mathbf{b}$$ and $$\mathbf{c}$$ and angle between $$\mathbf{b}$$ and $$\mathbf{c}$$ is $$\frac{\pi}{3}$$. Then, $$|\mathbf{a}+\mathbf{b}+\mathbf{c}|=$$

Let $$\mathbf{F}=2 \hat{i}+2 \hat{j}+5 \hat{k}, A=(1,2,5), B=(-1,-2,-3)$$ and $$\mathbf{B A} \times \mathbf{F}=4 \hat{i}+6 \hat{j}+2 \lambda \hat{k}$$, then $$\lambda=$$

$$O A B C$$ is a tetrahedron. If $$D, E$$ are the mid-points of $$O A$$ and $$B C$$ respectively, then $$\mathbf{D E}=$$

If $$\mathbf{a}+\mathbf{b}+\mathbf{c}=0$$ and $$|\mathbf{a}|=7,|\mathbf{b}|=5,|\mathbf{c}|=3$$ then the angle between $$\mathbf{b}$$ and $$\mathbf{c}$$ is

If $$P$$ and $$Q$$ are two points on the curve $$y=2^{x+2}$$ in the rectangular cartesian coordinate system such that $$\mathbf{O P} \cdot \hat{i}=-1, \mathrm{OQ} \cdot \hat{i}=2$$, then $$\mathrm{OQ}-4 \mathrm{OP}=$$

In quadrilateral $$A B C D, \mathbf{A B}=\mathbf{a}, \mathbf{B C}=\mathbf{b}$$. $$\mathbf{D A}=\mathbf{a}-\mathbf{b}, M$$ is the mid-point of $$B C$$ and $$X$$ is a point on DM such that, $$\mathbf{D X}=\frac{4}{5}$$ DM. Then, the points $$A, X$$ and $$C$$.

The vectors $$3 \mathbf{a}-5 \mathbf{b}$$ and $$2 \mathbf{a}+\mathbf{b}$$ are mutually perpendicular and the vectors $$a+4 b$$ and $$-\mathbf{a}+\mathbf{b}$$ are also mutually perpendicular, then the acute angle between $$\mathbf{a}$$ and $$\mathbf{b}$$ is

Let $$\mathbf{a}=x \hat{i}+y \hat{j}+z \hat{k}$$ and $$x=2 y$$. If $$|\mathbf{a}|=5 \sqrt{2}$$ and a makes an angle of $$135^{\circ}$$ with the Z-axis, then $$\mathbf{a}=$$

Let $$\mathbf{a}, \mathbf{b}, \mathbf{c}$$ be the position vectors of the vertices of a $$\triangle A B C$$. Through the vertices, lines are drawn parallel to the sides to form the $$\Delta A^{\prime} B^{\prime} C^{\prime}$$. Then, the centroid of $$\Delta A^{\prime} B^{\prime} C^{\prime}$$ is

The position vectors of the points $$A$$ and $$B$$ with respect to $$O$$ are $$2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$$ and $$2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$$. The length of the internal bisector of $$\angle B O A$$ of $$\triangle A O B$$ is (take proportionality constant is 2)

Let $$\mathbf{u}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{v}=-3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}$$ and $$\mathbf{w}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+4 \hat{\mathbf{k}}$$. Then which of the following statement is true?

If a = (1, 1, 0) and b = (1, 1, 1), then unit vector in the plane of a and b and perpendicular to a is

Let $$\mathbf{a}=\hat{\mathbf{i}}$$ and $$\mathbf{b}=\hat{\mathbf{j}}$$, the point of intersection of the lines $$\mathbf{r} \times \mathbf{a}=\mathbf{b} \times \mathbf{a}$$ and $$\mathbf{r} \times \mathbf{b}=\mathbf{a} \times \mathbf{b}$$ is

Which of the following vector is equally inclined with the coordinate axes?

If $$\hat{\mathbf{i}}+4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$$, and $$3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$$ are position vectors of $$A, B$$ and $$C$$ respectively and if $$D$$ and $$E$$ are mid points of sides $$B C$$ and $$A C$$, then $$\mathbf{D E}$$ is equal to

If $$\mathbf{a}$$ and $$\mathbf{b}$$ are two vectors such that $$\frac{\mathbf{a} \cdot \mathbf{b}}{|\mathbf{a}||\mathbf{b}|} < 0$$ and $$|\mathbf{a} \cdot \mathbf{b}|=|\mathbf{a} \times \mathbf{b}|$$ then the angle between the vectors $$\mathbf{a}$$ and $$\mathbf{b}$$ is

Let $$\mathbf{a}, \mathbf{b}$$ and $$\mathbf{c}$$ be three-unit vectors and $$\mathbf{a} \cdot \mathbf{b}=\mathbf{a} \cdot \mathbf{c}=0$$. If the angle between $$\mathbf{b}$$ and $$\mathbf{c}$$ is $$\frac{\pi}{3}$$. Then $$[\mathbf{a b c}]^2$$ is equal to

Let $$x$$ and $$y$$ are real numbers. If $$\mathbf{a}=(\sin x) \hat{\mathbf{i}}+(\sin y) \hat{\mathbf{j}}$$ and $$\mathbf{b}=(\cos x) \hat{\mathbf{i}}+(\cos y) \hat{\mathbf{j}}$$, then $$|\mathbf{a} \times \mathbf{b}|$$ is

A vector makes equal angles $$\alpha$$ with $$X$$ and $$Y$$-axis, and $$90 \Upsilon$$ with $$Z$$-axis. Then, $$\alpha$$ is equal to (c) 45Yand 135Y (d) $90 \mathrm{Y}$

Angle made by the position vector of the point (5, $$-$$4, $$-$$3) with the positive direction of X-axis is

If the volume of the parallelopiped formed by the vectors $$\hat{\mathbf{i}}+a \hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{j}}+a \hat{\mathbf{k}}$$ and $$a \hat{\mathbf{i}}+\hat{\mathbf{k}}$$ becomes minimum, then $$a$$ is equal to

If $$\mathbf{a}=\frac{3}{2} \hat{\mathbf{k}}$$ and $$\mathbf{b}=\frac{2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}}{2}$$, then angle between $$\mathbf{a}+\mathbf{b}$$ and $$\mathbf{a}-\mathbf{b}$$ is

Let $$\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$$ and $$\mathbf{c}=7 \hat{\mathbf{i}}+9 \hat{\mathbf{j}}+11 \hat{\mathbf{k}}$$, then the area of parallelogram having diagonals $$\mathbf{a}+\mathbf{b}$$ and $$\mathbf{b}+\mathbf{c}$$ is

If $$\mathbf{a}$$ and $$\mathbf{b}$$ are two vectors such that $$|\mathbf{a}|=2, |\mathbf{b}|=3$$ and $$\mathbf{a}+t \mathbf{b}$$ and $$\mathbf{a}-t \mathbf{b}$$ are perpendicular, where $$t$$ is a positive scalar, then