Vector Algebra · Mathematics · AP EAPCET

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 $\hat{\mathbf{f}}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{g}}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$, then the projection vector of $\hat{\mathrm{f}}$ on $\hat{\mathrm{g}}$ 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 $7 a^2+24 a=$

If $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-\hat{\mathbf{k}},-3 \hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ are the position vectors of three points, $A, B, C$ respectively, then $A, B, C$

AP EAPCET 2024 - 21th May Evening Shift

If $\mathbf{a}, \mathbf{b}, \mathbf{c}, \mathbf{d}$ are position vectors of 4 points such that $2 a+3 b+5 c-10 d=0$, then the ratio in which the line joining $c$ and $d$ divides the line segment joining $a$ and $\mathbf{b}$ is

AP EAPCET 2024 - 21th May Evening Shift

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are 3 vectors such that $|\mathbf{a}|=5,|\mathbf{b}|=8,|\mathbf{c}|=11$ and $\mathbf{a}+\mathbf{b}+\mathbf{c}=\mathbf{0}$, then the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$ is

AP EAPCET 2024 - 21th May Evening Shift

$\mathbf{a}=\alpha \hat{\mathbf{i}}+\beta \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \quad \mathbf{b}=\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ 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 $\alpha+\beta=$

$\mathbf{c}$ is a vector along the bisector of the internal angle between the vectors $\mathbf{a}=4 \hat{\mathbf{i}}+7 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}$ and $\mathbf{b}=12 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$. If the magnitude of $\mathbf{c}$ is $3 \sqrt{13}$, then c=

$\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ are two vectors and $\mathbf{c}$ is a unit vectors lying in the plane of $\mathbf{a}$ and $\mathbf{b}$. If $\mathbf{c}$ is perpendicular to $\mathbf{b}$, then $\mathbf{c}(\hat{\mathbf{i}}+\hat{\mathbf{j}}+2 \hat{\mathbf{k}})=$

If $\mathbf{a}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}, \mathbf{c}=2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-\hat{\mathbf{k}}$. $\mathbf{d}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ are four vector, then $(\mathbf{a} \times \mathbf{c}) \times(\mathbf{b} \times \mathbf{d})=$

The angle between the diagonals of the parallelogram whose adjacent sides are $2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}, \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ is

If the points having the position vectors $-i+4 j-4 k_{\text {, }}$, $3 i+2 j-5 k,-3 i+8 j-5 k$ and $-3 i+2 j+\lambda k$ are coplanar, then $\lambda=$

If $|f|=10,|g|=14$ and $|f-g|=15$, then $|f+g|=$

If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are three vectors such that $|\mathbf{a}|=|\mathbf{b}|=|\mathbf{c}|=\sqrt{3}$ and $(a+b-c)^2+(b+c-a)^2+(c+a-b)^2=36$, then $|2 a-3 b+2 c|=$

$\mathbf{a}, \mathbf{b}, \mathbf{c}$ are non-coplanar vectors. If $\alpha \mathbf{d}=\mathbf{a}+\mathbf{b}+\mathbf{c}$ and $\beta \mathbf{a}=\mathbf{b}+\mathbf{c}+\mathbf{d}$, then $|\mathbf{a}+\mathbf{b}+\mathbf{c}+\mathbf{d}|=$

AP EAPCET 2024 - 20th May Morning Shift

$\mathbf{u}, \mathbf{v}$ and $\mathbf{w}$ are three unit vectors. Let $\hat{\mathbf{p}}=\hat{\mathbf{u}}+\hat{\mathbf{v}}+\hat{\mathbf{w}} \cdot \hat{\mathbf{q}}=\hat{\mathbf{u}} \times(\hat{\mathbf{v}} \times \hat{\mathbf{w}})$. If $\hat{\mathbf{p}} \cdot \hat{\mathbf{u}}=\frac{3}{2} \cdot \hat{\mathbf{p}} \hat{\mathbf{v}}=\frac{7}{4}|\hat{\mathbf{p}}|=2$ and $v=K . q$, then $K=$

AP EAPCET 2024 - 20th May Morning Shift

If $\mathbf{a}$ and $\mathbf{b}$ are the two non collinear vectors, then $|\mathbf{b}|\mathbf{a}+|\mathbf{a}| \mathbf{b}$ represents

AP EAPCET 2024 - 20th May Morning Shift

If $L M N$ are the mid-points of the sides $P Q, Q R$ and $R P d$ $\triangle P Q R$ respectively, then $$ \begin{aligned} & \mathbf{Q M}+\mathbf{L N}+\mathbf{M L}+\mathbf{R N}-\mathbf{M N}-\mathbf{Q L}= \end{aligned} $$

Let $\mathbf{a} \times \mathbf{b}=7 \hat{\mathbf{i}}-5 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}$ and $\mathbf{a}=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$. If the length of projection of $\mathbf{b}$ on $\mathbf{a}$ is $$ \frac{8}{\sqrt{14}}, \text { then }|b|= $$

Let $A B C$ be an equilateral triangle of side a. $M$ and $N$ are two points on the sides $A B$ and $A C$, respectively such that $\mathbf{A N}={ }^{\prime} K \mathbf{A C}$ and $\mathbf{A B}=3 \mathbf{A M}$. If the vectors $\mathbf{B N}$ and $\mathbf{C M}$ are perpendicular, then $K=$

Let $\mathbf{a}$ and $\mathbf{b}$ be two non-collinear vector of unit modulus. If $\mathbf{u}=\mathbf{a}-(\mathbf{a} \cdot \mathbf{b}) \mathbf{b}$ and $\mathbf{v}=\mathbf{a} \times \mathbf{b}$, then $|\mathbf{v}|=$

In a regular hexagon $A B C D E F, \mathbf{A B}=\mathbf{a}$ and $\mathbf{B C}=\mathbf{b}$, then $F A=$

If $\mathbf{f}, \mathbf{g}, \mathbf{h}$ be mutually orthogonal vectors of equal magnitudes, then the angle between the vectors $\mathbf{f}+\mathbf{g}+\mathbf{h}$ and $\mathbf{h}$ is

Let $\mathbf{a}, \mathbf{b}$ be two unit vectors. If $\mathbf{c}=\mathbf{a}+2 \mathbf{b}$ and $\mathbf{d}=5 \mathbf{a}-4 \mathbf{b}$ are perpendicular to each other, then the angle between $a$ and $b$ is

If the vectors $\mathbf{a}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$, $\mathbf{c}=3 \hat{\mathbf{i}}+p \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ are coplanar, then $p=$

If $(\alpha, \beta, \gamma)$ are the direction cosines of an angular bisector of two lines whose direction ratios are $(2,2,1)$ and $(2,-1,-2)$, then $(\alpha+\beta+\gamma)^2=$

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 $$\lambda=$$

AP EAPCET 2022 - 4th July Evening Shift

$$O A B C$$ 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

AP EAPCET 2022 - 4th July Evening Shift

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 $$\mathrm{OQ}-4 \mathrm{OP}=$$

AP EAPCET 2022 - 4th July Evening Shift

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