Algebra Questions (626)

72. Let \(\vec{a} = \hat{i} + \hat{j} + \hat{k}\), \(\vec{b} = 4\hat{i} + 3\hat{j} + 4\hat{k}\) and \(\vec{c} = \hat{i} + \alpha\hat{j} + \beta\hat{k}\) are linearly dependent and \(|\vec{c}| = \sqrt{3}\), then \(|\alpha| + \beta =\) ________.

The area (in sq. units) of the parallelogram, whose diagonals are along the vectors \(8\hat{i}-6\hat{j}\) and \(3\hat{i}+4\hat{j}-12\hat{k}\), is

[JEE Main 2021] Let \(\vec{a}=2\hat{i}+\hat{j}-2\hat{k}\) and \(\vec{b}=\hat{i}+\hat{j}\). Let \(\vec{c}\) be a vector such that \(\vec{a}\cdot\vec{c}=|\vec{c}|\), \(|\vec{c}-\vec{a}|=2\sqrt2\) and the angle between \(\vec{a}\times\vec{b}\) and \(\vec{c}\) is \(\dfrac\pi6\). Then \(|(\vec{a}\times\vec{b})\times\vec{c}|\) equals

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 \(|\vec{b}|\).

Let \(\vec{a}=\hat{i}+\hat{j}+\hat{k}\), \(\vec{b}=2\hat{i}+\alpha\hat{j}+\hat{k}\), \(\vec{c}=\hat{i}-4\hat{j}+5\hat{k}\). If the volume of the parallelepiped with adjacent sides \(\vec{a},\vec{b},\vec{c}\) is 2, find the value of \(6\alpha\).

Let PQRS be a parallelogram whose diagonals \(\overrightarrow{PR}=3\hat{i}+\hat{j}\) and \(\overrightarrow{QS}=\hat{i}-\hat{j}+\hat{k}\). Find the area of PQRS.

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

Let \(\vec{a}=\hat{i}+2\hat{j}+3\hat{k}\) and \(\vec{b}=\hat{i}+\hat{j}-\hat{k}\). If \(\hat{c}\) is a unit vector perpendicular to both \(\vec{a}\) and \(\vec{b}\), find \(|\hat{c}\cdot(3\hat{i}-\hat{j}+\hat{k})|\).

Let \(\vec{a}=2\hat{i}+3\hat{j}-\hat{k}\), \(\vec{b}=\hat{i}-\hat{j}+2\hat{k}\), \(\vec{c}=2\hat{i}+\hat{j}-\hat{k}\). If \([\vec{a}+\vec{b},\,\vec{b}+\vec{c},\,\vec{c}+\vec{a}]=\lambda[\vec{a},\vec{b},\vec{c}]\), find \(\lambda\).

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?

Given three vectors \(\vec{U}=\hat{i}+\hat{j}+\hat{k}\), \(\vec{V}=\hat{i}+\hat{j}-\hat{k}\), \(\vec{W}=\hat{i}-\hat{j}+\hat{k}\). Which of the following hold?

Consider the regular hexagon ABCDEF with centre at O (origin). Find \(\vec{AD} + \vec{EB} + \vec{FC}\)

From the parallelogram law of forces, if \(R^2 = P^2 + Q^2 + 2PQ\cos\alpha\) and equations are set up for doubled forces, the ratio \(P^2 : Q^2 : R^2\) equals:

If ABCD is a parallelogram and the position vectors of A, B and C are $\vec{i} + 3\vec{j} + 5\vec{k}$, $\vec{i} + \vec{j} + \vec{k}$ and $7\vec{i} + 7\vec{j} + 7\vec{k}$, then the position vector of D will be

The horizontal force and the force inclined at an angle 60° with the vertical, whose resultant is in vertical direction of P kg, are

If the vectors \(6\mathbf{i}, 2\mathbf{j}\) and \(3\mathbf{i} + 6\mathbf{j} - 2\mathbf{k}\) form a triangle, determine the type of triangle.

Let a = 3i − 6j + k and b = 2i − 4j + lk. Since a and b are parallel, find the value of l.

If position vector of point A is a + 2b and a divides AB in the ratio 2 : 3, then the position vector of B is

If a and b are the position vectors of A and B respectively, then the position vector of a point C on AB produced such that AC = 3 AB is

We have a, b, c are non-zero vectors and \(\vec{a} \neq \lambda_1 \vec{b}\) and \(\vec{b} \neq \mu_1 \vec{c}\), \(\lambda_1, \mu_1 \neq 0\). Given:\(\vec{a} + 2\vec{b} = m\vec{c}\)     ...(1)and \(\vec{b} + 3\vec{c} = n\vec{a}\)     ...(2)where \(m, n \neq 0\) are reals. Find \(\vec{a} + 2\vec{b} + 6\vec{c}\) = ?

If \(a\), \(b\) and \(c\) are \(p\)th, \(q\)th, \(r\)th terms of HP and \(\vec{u} = (q - r)\vec{i} + (r - p)\vec{j} + (p - q)\vec{k}\), \(\vec{v} = \frac{1}{a}\vec{i} + \frac{1}{b}\vec{j} + \frac{1}{c}\vec{k}\), then

The angles of a triangle, two of whose sides are represented by vectors 3(a × b) and b - (a · b)a, where b is a non-zero vector and a is a unit vector in the direction of a, are

Direction of the ant's resultant displacement after first three steps is :

If the position vectors of the points \(A\), \(B\) and \(C\) be \(\vec{i} + \vec{j}\), \(\vec{i} - \vec{j}\) and \(a\vec{i} + b\vec{j} + c\vec{k}\) respectively, then the points \(A\), \(B\) and \(C\) are collinear, if

A vector \(\vec{C}\) directed along internal bisector of angle between vectors \(\vec{A} = 7\vec{i} - 4\vec{j} - 4\vec{k}\) and \(\vec{B} = -2\vec{i} - \vec{j} + 2\vec{k}\) with \(|\vec{C}| = 5\sqrt{6}\) is

The sine of the angle between the vectors a = 3i + j + k and b = 2i - 2j + k is

In a parallelogram ABCD, \(\vec{AB} = \vec{i} + \vec{j} + \vec{k}\) and diagonal \(\vec{AC} = \vec{i} - \vec{j} + \vec{k}\) and area of parallelogram is 8 sq units, then \(\angle BAC\) is equal to

Let $\vec{a}=\hat{i}-2\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$, $\vec{c}=\lambda\hat{i}+\hat{j}+\hat{k}$ and $\vec{v}=\vec{a}\times\vec{b}$. If $\vec{v}\cdot\vec{c}=11$ and the length of the projection of $\vec{b}$ on $\vec{c}$ is $p$, then $9p^2$ is equal to:

Let $\vec{u} = \hat{i} - \hat{j} - 2\hat{k}$, $\vec{v} = 2\hat{i} + \hat{j} - \hat{k}$, $\vec{v}\cdot\vec{w} = 2$ and $\vec{v}\times\vec{w} = \vec{u} + \lambda\vec{v}$. Then $\vec{u}\cdot\vec{w}$ is equal to

Let $\vec{\alpha} = 4\hat{i} + 3\hat{j} + 5\hat{k}$ and $\vec{\beta} = \hat{i} + 2\hat{j} - 4\hat{k}$. Let $\vec{\beta}_1$ be parallel to $\vec{\alpha}$ and $\vec{\beta}_2$ be perpendicular to $\vec{\alpha}$. If $\vec{\beta} = \vec{\beta}_1 + \vec{\beta}_2$, then the value of $5\vec{\beta}_2\cdot(\hat{i}+\hat{j}+\hat{k})$ is

Let $\vec{a} = \hat{i} + 2\hat{j} + \lambda\hat{k}$, $\vec{b} = 3\hat{i} - 5\hat{j} - \lambda\hat{k}$, $\vec{a}\cdot\vec{c} = 7$, $2\vec{b}\cdot\vec{c} + 43 = 0$, $\vec{a}\times\vec{c} = \vec{b}\times\vec{c}$. Then $|\vec{a}\cdot\vec{b}|$ is equal to

Let $\lambda \in \mathbb{R}$, $\vec{a} = \lambda\hat{i}+2\hat{j}-3\hat{k}$, $\vec{b} = \hat{i}-\lambda\hat{j}+2\hat{k}$. If $((\vec{a}+\vec{b})\times(\vec{a}\times\vec{b}))\times(\vec{a}-\vec{b}) = 8\hat{i}-40\hat{j}-24\hat{k}$, then $|\lambda(\vec{a}+\vec{b})\times(\vec{a}-\vec{b})|^2$ is equal to

Magnitude of the resultant displacement is given by :

Two adjacent sides of a parallelogram ABCD are given by \(\overrightarrow{AB} = 2\vec{i} + 10\vec{j} + 11\vec{k}\) and \(\overrightarrow{AD} = -\vec{i} + 2\vec{j} + 2\vec{k}\). The side AD is rotated by an acute angle \(\alpha\) in the plane of the parallelogram so that AD becomes \(\overrightarrow{AD'}\). If \(\overrightarrow{AD'}\) makes a right angle with the side AB, then the cosine of the angle \(\alpha\) is given by

Ex. 46 Statement I: If \(|\mathbf{a} + \mathbf{b}| = |\mathbf{a} - \mathbf{b}|\), then \(\mathbf{a}\) and \(\mathbf{b}\) are perpendicular to each other.Statement II: If the diagonals of a parallelogram are equal in magnitude, then the parallelogram is a rectangle.

Let a = \(2\mathbf{i} + \mathbf{j} - 2\mathbf{k}\) and b = \(\mathbf{i} + \mathbf{j}\). If c is a vector such that \(\mathbf{a} \cdot \mathbf{c} = |\mathbf{c}|\), \(|\mathbf{c} - \mathbf{a}| = 2\sqrt{2}\) and the angle between \(\mathbf{a} \times \mathbf{b}\) and c is 30°, then \(|(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}|\) is equal to

The non-zero vectors a, b and c are related by a = 8b and c = -7b. Find the angle between a and c.

If vectors $\vec{a} = \frac{\vec{i} + \vec{j}}{\sqrt{2}}, \vec{b} = \frac{-\vec{i} + \vec{j}}{\sqrt{2}}$ and $\vec{c} = \vec{k}$ then the value of $(\vec{r} \cdot \vec{a})^2 + (\vec{r} \cdot \vec{b})^2 + (\vec{r} \cdot \vec{c})^2$ is equal to:

Ex. 56 In the regular pentagon ABCDE with the given position vectors, AD divides EC in the ratio

If the volume of parallelepiped formed by the vectors a, b, c as three coterminous edges is 27 cu units, then the volume of the parallelepiped with \(\alpha = a + 2b - c\), \(\beta = a - b\) and \(\gamma = a - b - c\) as three coterminous edges is

The position vectors of the points A, B and C are i + 2j - k, i + j + k, and i + 3j + 2k, respectively. If A is [incomplete question]

If O be the circumcentre and O' be the orthocentre of the $\triangle ABC$, then $\vec{O'A} + \vec{O'B} + \vec{O'C}$ is equal to

A line segment has length 63 and direction ratios are 3, -2 and 6. Find the components of the line vector.

The value of \(\hat{i} \cdot (\hat{j} \times \hat{k}) + \hat{j} \cdot (\hat{k} \times \hat{i}) + \hat{k} \cdot (\hat{i} \times \hat{j})\) is

$(\vec{a} \times \vec{b}) \times \vec{c} = \vec{a} \times (\vec{b} \times \vec{c})$ is $\vec{a}$ and $\vec{c}$ are:

Let the cosine of angle between the vectors p and q be \(λ\) such that \(2\mathbf{p} + \mathbf{q} = \mathbf{i} + \mathbf{j}\) and \(\mathbf{p} + 2\mathbf{q} = \mathbf{i} - \mathbf{j}\), then \(λ\) is equal to

The position vector of the points which divides internally in the ratio 2 : 3 the join of the points 2a - 3b and 3a - 2b, is

Consider points \(A\), \(B\), \(C\) and \(D\) with position vectors \(7\hat{i} - 4\hat{j} + 7\hat{k}\), \(\hat{i} - 6\hat{j} + 10\hat{k}\), \(-\hat{i} - 3\hat{j} + 4\hat{k}\) and \(5\hat{i} - \hat{j} + \hat{k}\), respectively. Then \(ABCD\) is a

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

Let \(\vec{a} = \vec{i} + 2\vec{j} + \vec{k}\), \(\vec{b} = \vec{i} - \vec{j} + \vec{k}\), \(\vec{c} = \vec{i} + \vec{j} - \vec{k}\). A vector coplanar to \(\vec{a}\) and \(\vec{b}\) has a projection along \(\vec{c}\) of magnitude \(\frac{1}{3}\). Then the vector is: