Algebra Questions (626)

Coterminous edges $\vec{a},\vec{b},\vec{c}$ form parallelepiped of volume $V$. Volume with edges $\vec{a},\vec{b}+\vec{c},\vec{a}+2\vec{b}+3\vec{c}$ is equal to

For $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ with $10|a_i|<1$: (A) $\max|a_i|\leq|\vec{a}|$ and (B) $|\vec{a}|\leq3\max|a_i|$

Let $\lambda\in\mathbb{Z}$, $\vec{a}=\lambda\hat{i}+\hat{j}-\hat{k}$. If $|\vec{a}\times(2\hat{i}-\hat{j}+\hat{k})|^2+...$

$S$ = set of $(\lambda,\mu)$ for which $\lambda\hat{i}-\hat{j}+\hat{k}$, $\hat{i}+2\hat{j}+\mu\hat{k}$, $3\hat{i}-4\hat{j}+5\hat{k}$ coplanar, $\lambda-\mu=5$. Then $\sum_{S}80(\lambda^2+\mu^2)$ is equal to

$ABCD$ quadrilateral, $E,F$ midpoints of $AC,BD$. $(\overrightarrow{AB}-\overrightarrow{BC})+(\overrightarrow{AD}-\overrightarrow{DC})=k\overrightarrow{FE}$. Then $k$ is equal to

$|\vec{a}|=2$, $|\vec{b}|=3$, angle between $\vec{a},\vec{b}=\frac{\pi}{4}$. Then $|(\vec{a}+2\vec{b})\times(2\vec{a}-3\vec{b})|^2$ is equal to

Let $\vec{a}=6\hat{i}+\hat{j}-\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}$. If $\vec{c}$ is a vector such that $|\vec{c}|\geq6$, $\vec{a}\cdot\vec{c}=6|\vec{c}|$, $|\vec{c}-\vec{a}|=2\sqrt{2}$ and the angle between $\vec{a}\times\vec{b}$ and $\vec{c}$ is $60^\circ$, then $|(\vec{a}\times\vec{b})\times\vec{c}|$ is equal to:

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?

Let $\vec{a}=2\hat{i}+\alpha\hat{j}+\hat{k}$, $\vec{b}=-\hat{i}+\hat{k}$, $\vec{c}=\beta\hat{j}-\hat{k}$, where $\alpha$ and $\beta$ are integers and $\alpha\beta=-6$. Let the values of the ordered pair $(\alpha,\beta)$, for which the area of the parallelogram of diagonals $\vec{a}+\vec{b}$ and $\vec{b}+\vec{c}$ is $\dfrac{\sqrt{21}}{2}$, be $(\alpha_1,\beta_1)$ and $(\alpha_2,\beta_2)$. Then $\alpha_1^2+\beta_1^2-\alpha_2\beta_2$ is equal to

For $\lambda>0$, let $\theta$ be the angle between the vectors $\vec{a}=\hat{i}+\lambda\hat{j}-3\hat{k}$ and $\vec{b}=3\hat{i}-\hat{j}+2\hat{k}$. If the vectors $\vec{a}+\vec{b}$ and $\vec{a}-\vec{b}$ are mutually perpendicular, then the value of $(14\cos\theta)^2$ is equal to

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+3\hat{j}-5\hat{k}$ and $\vec{c}=3\hat{i}-\hat{j}+\lambda\hat{k}$ be three vectors. Let $\vec{r}$ be a unit vector along $\vec{b}+\vec{c}$. If $\vec{r}\cdot\vec{a}=3$, then $3\lambda$ is equal to:

Let \(\vec{a},\vec{b}\) be two non-zero non-collinear vectors. Then \(|\vec{a}\times\vec{b}|\) equals

Let \to ^ ^ ^ a = 3 i - j + 2k, \to ^ ^ b = a \times ( i - 2k) and \to ^ c = b \times k . Then the projection of \to ^ c - 2j \to is : on a

Let a = 2 ^i - ^j + 3 k^ , b = 3 ^i - 5 ^j + k^ and c be a vector such that a \times c = c \times b and ​ ​ ( a + c ) ⋅ ( b + c ) = 168. Then the maximum value of ∣ c ∣ 2 is :

The point having position vectors 2i + 3j + 4k, 3i + 4j + 2k and 4i + 2j + 3k are the vertices of

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{b}|=1$ and $|\vec{b}\times\vec{a}|=2$. Then $|(\vec{b}\times\vec{a})-\vec{b}|^2$ is equal to:

Let $\vec{a}=a_1\hat{i}+a_2\hat{j}+a_3\hat{k}$ and $\vec{b}=b_1\hat{i}+b_2\hat{j}+b_3\hat{k}$ be two vectors such that $|\vec{a}|=1$, $\vec{a}\cdot\vec{b}=2$ and $|\vec{b}|=4$. If $\vec{c}=2(\vec{a}\times\vec{b})-3\vec{b}$, then the angle between $\vec{b}$ and $\vec{c}$ is equal to:

Let $\vec{a}$ and $\vec{b}$ be two vectors such that $|\vec{a}|=1$, $|\vec{b}|=4$ and $\vec{a}\cdot\vec{b}=2$. If $\vec{c}=(2\vec{a}\times\vec{b})-3\vec{b}$ and the angle between $\vec{b}$ and $\vec{c}$ is $\alpha$, then $192\sin^2\alpha$ is equal to _____.

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=2\hat{i}+4\hat{j}-5\hat{k}$ and $\vec{c}=x\hat{i}+2\hat{j}+3\hat{k}$, $x\in\mathbb{R}$. If $\vec{d}$ is the unit vector in the direction of $\vec{b}+\vec{c}$ such that $\vec{a}\cdot\vec{d}=1$, then $(\vec{a}\times\vec{b})\cdot\vec{c}$ is equal to

Let $ABC$ be a triangle of area $15\sqrt{2}$ and the vectors $\overrightarrow{AB}=\hat{i}+2\hat{j}-7\hat{k}$, $\overrightarrow{BC}=a\hat{i}+b\hat{j}+c\hat{k}$ and $\overrightarrow{AC}=6\hat{i}+d\hat{j}-2\hat{k}$, $d>0$. Then the square of the length of the largest side of the triangle $ABC$ is _____

If $A(3,1,-1)$, $B\left(\dfrac{5}{3},\dfrac{7}{3},\dfrac{1}{3}\right)$, $C(2,2,1)$ and $D\left(\dfrac{10}{3},\dfrac{2}{3},-\dfrac{1}{3}\right)$ are the vertices of a quadrilateral $ABCD$, then its area is

The set of all $\alpha$, for which the vectors $\vec{a}=\alpha t\hat{i}+6\hat{j}-3\hat{k}$ and $\vec{b}=t\hat{i}-2\hat{j}-2\alpha t\hat{k}$ are inclined at an obtuse angle for all $t\in\mathbb{R}$, is

In an isosceles triangle \(ABC\) with \(AB=AC\), M is the midpoint of \(BC\). Given an equilateral triangle with side \(\vec{a}\), the scalar triple product \([\vec{a},\vec{b},\vec{c}]\) (where \(\vec{a},\vec{b},\vec{c}\) are sides) equals

Let \(f : R \to (0, 1)\) be a continuous function, then which of the following pair of vectors are linearly dependent for some \(x \in (0, 1)\)?

[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