Algebra Questions (626)

Given that $\vec{a}$ is perpendicular to $\vec{b}$ and p is a non-zero scalar, if $p\vec{r} + (\vec{r} \cdot \vec{b})\vec{a} = \vec{c}$ then $\vec{r} = $:

If $\vec{a}, \vec{b}, \vec{c}, \vec{d}$ are on a circle of radius $R$ whose centre is at origin and $\vec{c} - \vec{a}$ is perpendicular to $\vec{d} - \vec{b}$, then $|\vec{d} - \vec{a}|^2 + |\vec{b} - \vec{c}|^2$ (AC is diameter)

Let two non-collinear unit vectors \(\vec{a}\) and \(\vec{b}\) form an acute angle. A point P moves, so that at any time \(t\) the position vector \(\overrightarrow{OP}\) (where O is the origin) is given by \(\overrightarrow{OP} = \vec{a}\cos t + \vec{b}\sin t\). When P is farthest from origin O, let M be the length of \(\overrightarrow{OP}\) and \(\vec{u}\) be the unit vector along \(\overrightarrow{OP}\). Then find \(\vec{u}\) and \(M\).

If the position vectors of the points A and B are i + 3j - k and 3i - j - 3k, then what will be the position vector of the mid-point of AB?

The vector $\vec{OP} = 5\vec{i} + 12\vec{j} + 13\vec{k}$ turns through an angle of $\frac{\pi}{2}$ about $O$ passing through the positive side of $\vec{j}$ axis an iff way. The vector in the new position is:

If $\vec{a}, \vec{b}$ and $\vec{c}$ are three non-parallel unit vectors such that $\vec{a} \times (\vec{b} \times \vec{c}) = \frac{1}{2}\vec{b}$, the angle between $\vec{b}$ and $\vec{c}$ is:

Example 32. The length of longer diagonal of the parallelogram constructed on \(5\mathbf{a} + 2\mathbf{b}\) and \(\mathbf{a} - 3\mathbf{b}\), when it is given that \(|\mathbf{a}| = 2\sqrt{2}\), \(|\mathbf{b}| = 3\) and angle between \(\mathbf{a}\) and \(\mathbf{b}\) is \(\frac{\pi}{4}\), is

Let r, a, b and c be four non-zero vectors such that r · a = 0, |r × b| = |r| |b|, |r × c| = |r| |c|, then [a b c] is

Given \(\vec{\alpha} = (\lambda - 2)\vec{a} + \vec{b}\) and \(\vec{\beta} = (4\lambda - 2)\vec{a} + 3\vec{b}\). Vectors \(\vec{\alpha}\) and \(\vec{\beta}\) are collinear. Find \(\lambda\).

What should be added to vector a = 3i + 4j - 2k to get its resultant a unit vector i?

Let \(a, b, c\) denote the lengths of the sides of a triangle such that \((a-b)\vec{u} + (b-c)\vec{v} + (c-a)(\vec{u} \times \vec{v}) = 0\) for any two non-collinear vectors \(\vec{u}\) and \(\vec{v}\), then the triangle is

If O is origin and C is the mid-point of A(2, -1) and B(-4, 3), then the value of OC is

Let $OPQR$ is a tetrahedon such that O is origin and $\vec{p}, \vec{q}, \vec{r}$ are position vectors of P, Q, R respectively and $\alpha$ is the angle which OP makes with face PQR then:

Which of the following statements is/are correct? (a) If \(\mathbf{n} \cdot \mathbf{a} = 0, \mathbf{n} \cdot \mathbf{b} = 0\) and \(\mathbf{n} \cdot \mathbf{c} = 0\) for some non-zero vector \(\mathbf{n}\), then \([\mathbf{a} \mathbf{b} \mathbf{c}] = 0\)

Let \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) be distinct non-negative numbers. If the vectors \(a\vec{i} + a\vec{j} + c\vec{k}\), \(\vec{i} + \vec{k}\) and \(c\vec{i} + c\vec{j} + b\vec{k}\) lie in a plane, then \(c\) is

If the interior and exterior bisectors of the angle $A$ of a triangle $ABC$ meet the side $BC$ at $D$ and $E$, then :

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 \(\frac{AX}{XD}\) is

The direction cosines of vector a = 3i + 4j + 5k in the direction of positive axis of X, is

Ex. 64: Find the least positive integral value of x for which the angle between vectors \(\vec{a} = x\vec{i} - 3\vec{j} - \vec{k}\) and \(\vec{b} = 2x\vec{i} + x\vec{j} - \vec{k}\) is acute.

The resultant displacement of the ant after first two steps is:

The vector, directed along the internal bisector of the angle between the vectors $\vec{a} = 7\vec{i} - 4\vec{j} - 4\vec{k}$ & $\vec{b} = -2\vec{i} - \vec{j} + 2\vec{k}$ with $|\vec{c}| = 5\sqrt{6}$ is:

Ex. 54 The ratio \(PQ : DB\) is equal to

Ex. 106: Find AN + BP + CM where points M, N and P are taken on sides AB, BC and CA respectively, such that \frac{AM}{AB} = \frac{BN}{BC} = \frac{CP}{CA} = \alpha, with A at the origin, B and C having position vectors b and c respectively.

OP · OQ, where O is the origin (P is on the circle and curve, Q is inside the circle with integer abscissa)

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