Algebra Questions (626)

[JEE Main 2019] Let \(\vec{a}=\hat{i}-\hat{j}\) and \(\vec{b}=-\hat{i}+\hat{j}+\hat{k}\) be two given vectors. Let \(\vec{c}=\vec{a}\times\vec{b}\). Then which of the following is NOT true?

[JEE Main 2022] Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k}\) and \(\vec{b}=2\hat{i}+\hat{j}-\hat{k}\). If \(\vec{p}\) is such that \(\vec{a}\times\vec{p}=\vec{b}\) and \(\vec{a}\cdot\vec{p}=0\), then the value of \([\vec{a},\vec{p},\vec{b}]\) is

Let \(\hat{a}_1,\hat{a}_2,\ldots,\hat{a}_n\) be unit vectors in the plane such that \(\displaystyle\sum_{i=1}^{n}\hat{a}_i=\vec{0}\). Let \(k=\displaystyle\sum_{i

For any vector \(\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}\) with \(10|\vec{a}|\geq1\), consider: (A) \(|\vec{a}\times\hat{i}|^2+|\vec{a}\times\hat{j}|^2+|\vec{a}\times\hat{k}|^2=2|\vec{a}|^2\). (B) \(|\vec{a}\cdot\hat{i}|^2+|\vec{a}\cdot\hat{j}|^2+|\vec{a}\cdot\hat{k}|^2=|\vec{a}|^2\). Which is true?

[JEE Main 2020] The volume of a parallelepiped whose coterminous edges are \(\vec{u}=\hat{i}+\hat{j}+\lambda\hat{k}\), \(\vec{v}=\hat{i}+\hat{j}+3\hat{k}\), \(\vec{w}=2\hat{i}+\hat{j}+\hat{k}\) is 1 cubic unit. If \(\theta\) is the angle between the edges \(\vec{u}\) and \(\vec{w}\), then \(\cos\theta\) can be

Given a parallelogram \(OACB\) with \(\overrightarrow{OA}=\vec{a}\) and \(\overrightarrow{OB}=\vec{b}\). The length of the diagonal \(\overrightarrow{OC}\) is \(a\), \(b\), \(\sqrt{a^2+b^2-ab}\) or \(\sqrt{a^2+b^2+ab}\). If angle between \(\vec{a}\) and \(\vec{b}\) is \(\frac{2\pi}{3}\), then \(|\vec{a}+\vec{b}|\) is

If \([\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}] = \lambda[\vec{a} \; \vec{b} \; \vec{c}]^2\), then \(\lambda\) is equal to

It is given that \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) are linearly dependent, where \(\vec{a} = \hat{i}+\hat{j}+\hat{k}\), \(\vec{b} = 4\hat{i}+3\hat{j}+4\hat{k}\), \(\vec{c} = \hat{i}+\alpha\hat{j}+\beta\hat{k}\). Find \(\beta\).

If \([\vec{a}+2\vec{b}+3\vec{c}\quad \vec{b}+2\vec{c}+3\vec{a}\quad \vec{c}+2\vec{a}+3\vec{b}] = 36\), where \(\vec{a},\vec{b}\) and \(\vec{c}\) are three vectors, then the value of \(\begin{vmatrix} \vec{a}\cdot\vec{a} & \vec{a}\cdot\vec{b} & \vec{a}\cdot\vec{c} \\ \vec{a}\cdot\vec{b} & \vec{b}\cdot\vec{b} & \vec{b}\cdot\vec{c} \\ \vec{a}\cdot\vec{c} & \vec{c}\cdot\vec{b} & \vec{c}\cdot\vec{c} \end{vmatrix}\) is ________.

If \(α\) and \(β\) are two mutually perpendicular unit vectors and \(\{rα + rβ + s(α \times β)\}\), \([α + (α \times β)]\) and \(\{sα + sβ + t(α \times β)\}\) are coplanar, then \(s\) is equal to

Given $\vec{i} + \vec{j} + \vec{k}$. Two given lines intersected, if $7\vec{i} + 10\vec{j} + 13\vec{k} + s(2\vec{i} + 3\vec{j} + 4\vec{k}) = 3\vec{i} + 5\vec{j} + 7\vec{k} + t(\vec{i} + 2\vec{j} + 3\vec{k})$

Given $|\vec{a}| = 1$, $|\vec{b}| = 2$, $|\vec{c}| = 3$ and $\vec{a} \cdot \vec{b} = 0$, $\vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a}$ (as the three vectors are mutually perpendicular)

Ex. 52 In parallelogram ABCD, L is a point on BC which divides BC in the ratio 1 : 2. AL intersects BD at P. Point P divides AL in the ratio

Ex. 110: If AP, BQ and CR are the altitudes of acute \triangle ABC and 9\overrightarrow{AP} + 4\overrightarrow{BQ} + 7\overrightarrow{CR} = \mathbf{0}, then \angle ABC is equal to

Ex. 55 Let A, B, C, D, E represent vertices of a regular pentagon ABCDE with position vectors \(\vec{a}, \vec{a} + \vec{b}, \vec{b}, \lambda \vec{a}, \lambda \vec{b}\) respectively. The ratio \(\frac{AD}{BC}\) is equal to

Ex. 45 Statement I: If \(|\mathbf{a}| = 3\), \(|\mathbf{b}| = 4\) and \(|\mathbf{a} + \mathbf{b}| = 5\), then \(|\mathbf{a} - \mathbf{b}| = 5\).Statement II: The length of the diagonals of a rectangle is the same.

68. Let \(\sqrt{3}\hat{i} + \hat{j}\), \(\hat{i} + \sqrt{3}\hat{j}\) and \(\beta\hat{i} + (1-\beta)\hat{j}\) respectively be the position vectors of the points A, B and C with respect to the origin O. If the distance of C from the bisector of the acute angle between OA and OB is \(3/\sqrt{2}\), then the sum of all possible values of \(\beta\) is ________.

Let \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k}\). A vector \(\vec{b}\) satisfies \(\vec{a}\cdot\vec{b}=|\vec{b}|^2\) and \(|\vec{a}-\vec{b}|^2=7\). Find \(|\vec{b}\times\vec{a}|^2\).

The value(s) of \(x\) for which the angle between vectors \(\vec{a} = (1, x^2, 9)\) and \(\vec{b} = (4, 4x-2, 2)\) is such that \(\cos\theta = \dfrac{4(1)+(4x-2)(x)+6}{\sqrt{1+x^2+9}\cdot\sqrt{16+(4x-2)^2+4}}\) and \(\sqrt{(4)^2+(4x-2)^2+4} = 2\sqrt{1+x^2+9}\), find the possible values of \(x\).

Work done = \((\vec{F_1} + \vec{F_2}) \cdot \overrightarrow{AB}\) where \(\overrightarrow{OA} = \hat{i} + 2\hat{j} + 3\hat{k}\), \(\overrightarrow{OB} = 5\hat{i} + 4\hat{j} + \hat{k}\) and \(\vec{F_1} + \vec{F_2} = 7\hat{i} + 2\hat{j} - 4\hat{k}\). Find the work done.

Let r be perpendicular to a + b + c, where r = l[a b c]a + m[c a b]b + n[a b c]c and [a b c] = 2. Find the value of l + m + n.

The points A(2-x, 2, 2), B(2, 2-y, 2), C(2, 2, 2-z) and D(1, 1, 1) are coplanar. Find the locus of P(x, y, z).

The projections of a vector on coordinate axes are \(x_2 - x_1,\ y_2 - y_1,\ z_2 - z_1\). Also, \(x_2 - x_1 = 6,\ y_2 - y_1 = -3,\ z_2 - z_1 = 2\). Find the direction cosines of the vector.

Let \(\vec{a} = \hat{i} - \hat{k}\), \(\vec{b} = x\hat{i} + \hat{j} + (1-x)\hat{k}\) and \(\vec{c} = y\hat{i} + x\hat{j} + (1+x-y)\hat{k}\). Then \([\vec{a}, \vec{b}, \vec{c}]\) depends on

It is given that \(\vec{a}\), \(\vec{b}\) and \(\vec{c}\) are three unit vectors such that \[\vec{a} \times (\vec{b} \times \vec{c}) = \frac{\sqrt{3}}{2}(\vec{b}+\vec{c})\] Using this condition, find the angle between \(\vec{a}\) and \(\vec{b}\) (or relevant quantity as per full question).

The unit vector perpendicular to the vectors a = 6i + 2j + 3k and b = 3i - 6j - 2k is

The values of \(a\), for which the points \(A\), \(B\), \(C\) with position vectors \(2\hat{i} - \hat{j} + \hat{k}\), \(\hat{i} - 3\hat{j} - 5\hat{k}\) and \(a\hat{i} - 3\hat{j} + \hat{k}\), respectively, are the vertices of a right-angled triangle with \(C = \pi/2\) are

The number of integral values of $a$ for which the equation $x^4 - (a+2)x^3 + 2ax^2 + 4(a-2)x - 16 = 0$ has at least two positive roots; $a \in [-10, 10]$ is/are

In a parallelogram ABCD, \(|\overrightarrow{AB}| = a\), \(|\overrightarrow{AD}| = b\) and \(|\overrightarrow{AC}| = c\), then \(\overrightarrow{DB} \cdot \overrightarrow{AB}\) has the value

We have \(|\hat{x}+\hat{y}|^2 + |\hat{y}+\hat{z}|^2 + |\hat{z}+\hat{x}|^2\) where \(\hat{x}, \hat{y}, \hat{z}\) are unit vectors making angles \(\alpha, \beta, \gamma\) with each other. Find the minimum value of \(|\hat{x}+\hat{y}|^2 + |\hat{y}+\hat{z}|^2 + |\hat{z}+\hat{x}|^2\).

In the following figure, \(AB\), \(DE\) and \(GF\) are parallel to each other and \(AD\), \(BG\) and \(EF\) are parallel to each other. If \(CD : CE = CG : CB = 2 : 1\), then the value of area \((\triangle AEG)\) : area \((\triangle ABD)\) is equal to

If a and b are two unit vectors, i.e., |a| = |b| = 1 and angle between them is \(\frac{\pi}{3}\), then \(16[\mathbf{a} \mathbf{b} + \mathbf{a} \times \mathbf{b} \mathbf{b}]\) is equal to:

If $\vec{a} = \hat{i}+2\hat{k}$, $\vec{b} = \hat{i}+\hat{j}+\hat{k}$, $\vec{c} = 7\hat{i}-3\hat{k}+4\hat{k}$, $\vec{r}\times\vec{b}+\vec{b}\times\vec{c} = \vec{0}$ and $\vec{r}\cdot\vec{a}=0$, then $\vec{r}\cdot\vec{c}$ is equal to

If $ABC$ be a triangle of sides $a, b, c$ with position vectors of $A, B, C$ as $\vec{a}, \vec{b}$ and $\vec{c}$ respectively, then the position vector of its incentre is:

Let $\vec{a}=2\hat{i}+3\hat{j}+4\hat{k}$, $\vec{b}=\hat{i}-2\hat{j}-2\hat{k}$, $\vec{c}=-\hat{i}+4\hat{j}+3\hat{k}$. If $\vec{d}\perp\vec{b},\vec{c}$ and $\vec{a}\cdot\vec{d}=18$, then $|\vec{a}\times\vec{d}|^2$ is equal to

If $\vec{a}$ is a unit vector and projection of $\vec{x}$ along $\vec{a}$ is 2 units and $(\vec{a} \times \vec{x}) \cdot \vec{b} = \vec{x}$, then $\vec{x}$ is given by:

If the vectors $\vec{a} = \lambda\hat{i} + \mu\hat{j} + 4\hat{k}$, $\vec{b} = 2\hat{i} + 4\hat{j} - 2\hat{k}$ and $\vec{c} = 2\hat{i} + 3\hat{j} + \hat{k}$ are coplanar and the projection of $\vec{a}$ on the vector $\vec{b}$ is $\sqrt{54}$ units, then the sum of all possible values of $\lambda + \mu$ is equal to

$\vec{a}=\hat{i}+4\hat{j}+2\hat{k}$, $\vec{b}=3\hat{i}-2\hat{j}+7\hat{k}$, $\vec{c}=2\hat{i}-\hat{j}+4\hat{k}$. $\vec{d}\times\vec{b}=\vec{c}\times\vec{b}$, $\vec{d}\cdot\vec{a}=24$. Then $|\vec{d}|^2$ is equal to

The foot of perpendicular from the origin O to a plane P which meets the co-ordinate axes at the points A, B, C is $(2, a, 4)$, $a \in \mathbb{N}$. If the volume of the tetrahedron OABC is 144 unit$^3$, then which of the following points is NOT on P?

If three coterminous edges of a tetrahedron are $\vec{a}, \vec{b}, \vec{c}$ such that $|\vec{a}| = 2, |\vec{b}| = 3, |\vec{c}| = 4$, angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{3}$, $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{4}$ and $\vec{c}$ and $\vec{a}$ is $\frac{\pi}{6}$. The area of the base is $2$ sq. units, then the height of the tetrahedron is:

If $\vec{\alpha}$ and $\vec{\beta}$ be two perpendicular unit vectors such that $\vec{x} = \vec{\beta} - (\vec{a} \times \vec{x})$, then $|\vec{x}|$ is equal to:

Let \(\vec{a}=\hat{i}+\hat{j}+2\hat{k}\), \(\vec{b}=2\hat{i}-\hat{j}+\hat{k}\), \(\vec{c}=3\hat{i}-\hat{k}\). A vector \(\vec{v}\) is coplanar with \(\vec{a}\) and \(\vec{b}\), perpendicular to \(\vec{c}\), and satisfies \(\vec{v}\cdot(\hat{i}+2\hat{j}+\hat{k})=8\). Find \(|\vec{v}|^2\).

Let $\vec{a} = a\vec{i} + b\vec{j} + c\vec{k}$ and $\vec{\beta} = b\vec{i} + \vec{c}\vec{j} + a\vec{k}$, where $a, b, c \in \mathbb{R}$. If '$\theta$' be the angle between $\vec{a}$ and $\vec{\beta}$ then:

Let \(OAB\) be a regular triangle (equilateral) with \(O\) at the origin. If \(\overrightarrow{OA}=\vec{a}\) and \(\overrightarrow{OB}=\vec{b}\), then which of the following hold?

A vector $\vec{a} = a\vec{i} + b\vec{j} + c\vec{k}$ is said to be rational vector. If $a, b, c$ are all rational. If a rational vector with magnitude as positive integer makes an angle $\pi/4$ with vector $\vec{b} = \sqrt{2}\vec{i} + \sqrt{2}\vec{j} + \vec{k}$, then $\vec{a}$:

$ABC$ is a triangle. $AD, AD'$ are internal and external bisectors of angle $A$, meeting $BC$ at $D$ and $D'$ respectively. $A'$ is the mid-point of $DD'$ and $B', C'$ are similar points on $CA$ and $AB$. Then $A', B', C'$:

Let $\vec{a} = \vec{i} + \vec{j}$ and $\vec{b} = 2\vec{i} - \vec{k}$. The point of intersection of the lines $\vec{r} \times \vec{a} = \vec{b} \times \vec{a}$ and $\vec{r} \times \vec{b} = \vec{a} \times \vec{b}$ is:

If for a plane, $x$, $y$, $z$ intercepts are $8$, $4$ respectively, then the length of the perpendicular from the origin on to the plane is

If $\vec{a},\vec{b},\vec{c},\vec{d}$ are coplanar, then $[\vec{a}\vec{b}\vec{c}]$ is equal to

If $(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) \cdot (\vec{c} \times \vec{b}) = 0$ then which of the following is always true: