Algebra Questions (626)

Let position vectors of points A, B and C of triangle ABC respectively be \(\vec{i} + \vec{j} + 2\vec{k}\), \(\vec{i} + 2\vec{j} + \vec{k}\) and \(2\vec{i} + \vec{j} + \vec{k}\). Let \(l_1\), \(l_2\) and \(l_3\) be the lengths of perpendiculars drawn from the orthocenter 'O' on the sides AB, BC and CA, then \((l_1 + l_2 + l_3)\) equals

Let a vector $\vec{a}=\sqrt{2}\hat{i}-\hat{j}+\lambda\hat{k}$, $\lambda>0$, make an obtuse angle with the vector $\vec{b}=-\lambda^2\hat{i}+4\sqrt{2}\hat{j}+4\sqrt{2}\hat{k}$ and an angle $\theta$, $\dfrac{\pi}{6}<\theta<\dfrac{\pi}{2}$, with the positive $z$-axis. If the set of all possible values of $\lambda$ is $(\alpha,\beta)-\{\gamma\}$, then $\alpha+\beta+\gamma$ is equal to ___.

\((\vec{a} \cdot \vec{i})\vec{i} + (\vec{a} \cdot \vec{j})\vec{j} + (\vec{a} \cdot \vec{k})\vec{k}\) is equal to

If a + b + c = 0 and |a| = 3, |b| = 5, |c| = 7, then the angle between a and b is

The direction cosines of the vector 3i - 4j + 5k are

In a triangle AOB, R and Q are the points on the side OB and AB respectively such that \(3OR = 2RB\) and \(2AQ = 3QB\). Let OQ and AR intersect at the point P (where O is origin). If the point P divides OQ in the ratio of m:1, then m is:

A vector equally inclined to the vectors \(\mathbf{i} - \mathbf{j} + \mathbf{k}\) and \(\mathbf{i} + \mathbf{j} - \mathbf{k}\) then the plane containing them is

B divided AC in ratio

Let $\vec{u}$ and $\vec{v}$ be unit vectors. if $\vec{w}$ is a vector such that $\vec{w} + (\vec{w} \times \vec{u}) = \vec{v}$, if the maximum volume of the parallelepiped formed by $\vec{u}, \vec{v}$ and $\vec{w}$ is $p$ then $12p = _______.

Given a parallelogram ABCD. If |AB| = a, |AD| = b and |AC| = c, then DB · AB has the value

Five points given by A, B, C, D and E are in a plane. Three forces $\vec{AC}$, $\vec{AD}$ and $\vec{AE}$ act at A and three forces $\vec{CB}$, $\vec{DB}$ and $\vec{EB}$ act at B. Then, their resultant is

If the lines $\vec{r} = \vec{a} + (\vec{b} \times \vec{c})$, and $\vec{r} = \vec{b} + s(\vec{c} \times \vec{a})$ intersect (t and s are scalars) then:

Ex. 99 Which of the following is true?

Let $A(2,3,5)$ and $C(-3,4,-2)$ be opposite vertices of a parallelogram $ABCD$. If the diagonal $\overrightarrow{BD}=\hat{i}+2\hat{j}+3\hat{k}$, then the area of the parallelogram is equal to:

[JEE Main 2021] Let \(\vec{a},\vec{b},\vec{c}\) be three mutually perpendicular unit vectors. The angle \(\theta\) between each of them and the vector \(\vec{a}+\vec{b}+\vec{c}\) is

If OABC is a tetrahedron such that $OA^2 + BC^2 = OB^2 + CA^2 = OC^2 + AB^2$, then which of the following is not true?

Let \(\vec{a} = \vec{i} + 2\vec{j} - 3\vec{k}\) and \(\vec{b} = 2\vec{i} - 3\vec{j} + 5\vec{k}\). If \(\vec{r} \times \vec{a} = \vec{b} \times \vec{r}\), \(\vec{r} \times (a\vec{i} + 2\vec{j} + \vec{k}) = 3\) and \(\vec{r} \times (2\vec{i} + 5\vec{j} - a\vec{k}) = -1\), where \(a \in \mathbb{R}\), then the value of \(a + |\vec{r}|^2\) is equal to

If b and c are orthogonal unit vectors and \(\mathbf{b} \times \mathbf{c} = \mathbf{a}\), then \([\mathbf{a} + \mathbf{b} + \mathbf{c} \mathbf{a} + \mathbf{b} \mathbf{b} + \mathbf{c}]\) is equal to:

$p_1 + p_2$ is equal to:

If the non-zero vectors $\vec{a}$ and $\vec{b}$ are perpendiculars to each other then the solution of the equation $\vec{r} \times \vec{a} = \vec{b}$ is given by:

The position vectors of vertices of ∆ABC are \(a\), \(b\), \(c\) and \(a \cdot a = b \cdot b = c \cdot c = 3\). If \([a b c] = 0\), then the position vector of the orthocentre of ∆ABC is

A unit vector perpendicular to the vector $\mathbf{i} + 2\mathbf{j} + 2\mathbf{k}$ and making equal angles with X and Y-axes can be:

If $4\vec{a} + 5\vec{b} + 9\vec{c} = \vec{0}$, then $(\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a})$ is equal to:

Let three vectors \(\vec{a},\vec{b},\vec{c}\) satisfy \(\vec{a}\times\vec{b}=\vec{c}\) and \(\vec{b}\times\vec{c}=\vec{a}\), with \(|\vec{a}|=1\). If the angle between \(\vec{b}\) and \(\vec{c}\) is \(\dfrac{\pi}{6}\), find \(|\vec{b}|\).

If a and b are the vectors determined by two adjacent sides of a regular hexagon, then vector EO is

Let OABCD be a pentagon in which the sides OA and CB are parallel and the sides OD and AB are parallel. Also, OA : CB = 2 : 1 and OD : AB = 1 : 3. The ratio \(\frac{OX}{XC}\) is

A vector whose modulus is \(\sqrt{51}\) and makes the same angle with a = \(\frac{\mathbf{i} - 2\mathbf{j} + 2\mathbf{k}}{3}\), b = \(\frac{-4\mathbf{i} - 3\mathbf{k}}{5}\) and c = j, will be

If the ratio of area of quadrilateral PQBR and area of △OPA is \frac{a}{b}, then find (b − a) where a and b are coprime numbers.

The 3-dimensional vectors v1, v2, v3 satisfying \(\mathbf{v}_1 \cdot \mathbf{v}_1 = 4\), \(\mathbf{v}_1 \cdot \mathbf{v}_2 = -2\), \(\mathbf{v}_1 \cdot \mathbf{v}_3 = 6\), \(\mathbf{v}_2 \cdot \mathbf{v}_2 = 2\), \(\mathbf{v}_2 \cdot \mathbf{v}_3 = -5\), \(\mathbf{v}_3 \cdot \mathbf{v}_3 = 29\), then v3 may be

Let \(\vec{a}=2\hat{i}-\hat{j}+4\hat{k}\) and \(\vec{b}=\hat{i}+\alpha\hat{j}+\beta\hat{k}\). If \(\vec{b}\) is perpendicular to \(3\hat{i}-4\hat{j}+\hat{k}\) and the projection of \(\vec{b}\) on \(\vec{a}\) is \(\dfrac{17}{\sqrt{21}}\), find \(|\vec{b}|\).

A(2, 6, 2), B(-4, 0, $\lambda$), C(2, 3, -1) and D(4, 5, 0), $|\lambda| \leq 5$ are the vertices of a quadrilateral ABCD. If its area is 18 square units, then $5-6\lambda$ is equal to _____.

Let $\vec{v} = \alpha\hat{i}+2\hat{j}-3\hat{k}$, $\vec{w} = 2\alpha\hat{i}+\hat{j}-\hat{k}$, and $\vec{u}$ be a vector such that $|\vec{u}|=\alpha>0$. If the minimum value of the scalar triple product $[\vec{u}\,\vec{v}\,\vec{w}]$ is $-\alpha\sqrt{3401}$, and $|\vec{u}\cdot\hat{i}|^2 = \frac{m}{n}$ where m and n are coprime natural numbers, then $m+n$ is equal to _____.

Two given points $P$ and $Q$ in the rectangular cartesian coordinates lie on $y = 2^{x^2}$ such that $\overrightarrow{OP} \cdot \hat{i} = -1$ and $\overrightarrow{OQ} \cdot \hat{i} = +2$ where $\hat{i}$ is a unit vector along the x-axis. The magnitude of $\frac{\overrightarrow{OQ} - 4\overrightarrow{OP}}{2}$ is _______.

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=\sqrt{14}$, $|\vec{b}|=\sqrt{6}$ and $|\vec{a}\times\vec{b}|=\sqrt{48}$. Then $(\vec{a}\cdot\vec{b})^2$ is equal to _____.

For two particular vectors A and B, it is known that A × B = B × A. What must be true about the two vectors?

Arc $PQ$ subtends right angle at centre $O$. Midpoint $R$ of arc. $\overrightarrow{OP}=\vec{u}$, $\overrightarrow{OR}=\vec{v}$, $\overrightarrow{OQ}=\alpha\vec{u}+\beta\vec{v}$. Then $\alpha,\beta^2$ satisfy

$\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=\hat{i}+\hat{j}-\hat{k}$. $\vec{c}$: $\vec{a}\cdot\vec{c}=11$, $\vec{b}\cdot(\vec{a}\times\vec{c})=27$, $\vec{b}\cdot\vec{c}=-\sqrt{3}|\vec{b}|$. Then $|\vec{a}\times\vec{c}|^2$ is equal to

Let $\vec{a} = 2\hat{i}+\hat{j}+\hat{k}$, and $\vec{b}$ and $\vec{c}$ be two nonzero vectors such that $|\vec{a}+\vec{b}+\vec{c}| = |\vec{a}+\vec{b}-\vec{c}|$ and $\vec{b}\cdot\vec{c}=0$. Consider the following two statements: (A) $|\vec{a}+\lambda\vec{c}| \geq |\vec{a}|$ for all $\lambda \in \mathbb{R}$. (B) $\vec{a}$ and $\vec{c}$ are always parallel. Then:

Let $\vec{a}$ and $\vec{b}$ be two vectors. Let $|\vec{a}|=1$, $|\vec{b}|=4$ and $\vec{a}\cdot\vec{b}=2$. If $\vec{c} = (2\vec{a}\times\vec{b})-3\vec{b}$, then the value of $\vec{b}\cdot\vec{c}$ is

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three non-zero non-coplanar vectors. Let the position vectors of four points A, B, C and D be $\vec{a}-\vec{b}+\vec{c}$, $\lambda\vec{a}-3\vec{b}+4\vec{c}$, $-\vec{a}+2\vec{b}-3\vec{c}$ and $2\vec{a}-4\vec{b}+6\vec{c}$ respectively. If $\overrightarrow{AB}$, $\overrightarrow{AC}$ and $\overrightarrow{AD}$ are coplanar, then $\lambda$ is:

Let a, b and c be three vectors that \([\vec{a}\;\vec{b}\;\vec{c}] = 2\). If \(\vec{r} = l(\vec{b}\times\vec{c}) + m(\vec{c}\times\vec{a}) + n(\vec{a}\times\vec{b})\) be perpendicular to \(\vec{a}+\vec{b}+\vec{c}\), then the value of \(l+m+n\) is ________.

If $|\vec{a}| = |\vec{b}| = |\vec{c}| = 2$ and $\vec{a} \cdot \vec{b} = \vec{b} \cdot \vec{c} = \vec{c} \cdot \vec{a} = -1$, then $|\vec{a} \times \vec{b} \times \vec{c} \times \vec{a}|$ is_______.

Sum of all $\alpha$ for which $\hat{i}-2\hat{j}+3\hat{k}$, $2\hat{i}-3\hat{j}+4\hat{k}$, $(\alpha+1)\hat{i}+2\hat{k}$, $9\hat{i}+(\alpha-8)\hat{j}+6\hat{k}$ are coplanar, is equal to

Let $P$ be a point in the plane of the vectors $\overrightarrow{AB}=3\hat{i}+\hat{j}-\hat{k}$ and $\overrightarrow{AC}=\hat{i}-\hat{j}+3\hat{k}$ such that $P$ is equidistant from the lines AB and AC. If $|\overrightarrow{AP}|=\dfrac{\sqrt{5}}{2}$, then the area of the triangle ABP is:

Let $\vec{a}$, $\vec{b}$, $\vec{c}$ be three vectors such that $|\vec{a}|=\sqrt{31}$, $4|\vec{b}|=|\vec{c}|=2$ and $2(\vec{a}\times\vec{b}) = 3(\vec{c}\times\vec{a})$. If the angle between $\vec{b}$ and $\vec{c}$ is $\frac{2\pi}{3}$, then $\left(\frac{\vec{a}\times\vec{c}}{\vec{a}\cdot\vec{b}}\right)^2$ is equal to _____.

In a trapezium, the vector \(\vec{BC} = \lambda \vec{AD}\). We will then find that \(\vec{p} = \vec{AC} + \vec{BD}\) is collinear with \(\vec{AD}\). If \(\vec{p} = \mu \vec{AD}\), then find the relationship between \(\lambda\) and \(\mu\).

Given that the vectors $\vec{a}, \vec{b}$ and $\vec{c}$ (no two of them are collinear). Further if $(\vec{a} + \vec{b})$ is collinear with $\vec{c}, (\vec{b} + \vec{c})$ is collinear with $\vec{a}$ and $|\vec{a}| = |\vec{b}| = |\vec{c}| = \sqrt{2}$. Then the value of $|\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{c} + \vec{c} \cdot \vec{a}|$ is ______.

Let $\vec{u}, \vec{v}, \vec{w}$ be such that $|\vec{u}| = 1$, $|\vec{v}| = 2$, $|\vec{w}| = 3$. If the projection $\vec{v}$ along $\vec{u}$ is equal to that of $\vec{w}$ along $\vec{u}$ and $\vec{v}, \vec{w}$ are perpendicular to each other, then $\frac{|\vec{u} - \vec{v}|^2}{2}$ equals _______.

We have \(|2\vec{a} - \vec{b}|^2 = 25 \Rightarrow 4a^2 + b^2 - 4\vec{a}\cdot\vec{b} = 25\). Given \(|\vec{a}| = 2, |\vec{b}| = 3\), find \(|2\vec{a} + \vec{b}|^2\):

A particle is acted upon by constant forces \(4\hat{i}+\hat{j}-3\hat{k}\) and \(3\hat{i}+\hat{j}-\hat{k}\) which displace it from a point \(\hat{i}+2\hat{j}+3\hat{k}\) to the point \(5\hat{i}+4\hat{j}+\hat{k}\). The work done in standard units by the forces is given by