Vector Algebra Questions (573)

The vector $\vec{a} = -\hat{i} + 2\hat{j} + \hat{k}$ is rotated through a right angle, passing through the y-axis in its way and the resulting vector is $\vec{b}$. Then the projection of $3\vec{a} + \sqrt{2}\vec{b}$ on $\vec{c} = 5\hat{i} + 4\hat{j} + 3\hat{k}$ is

Let O be the origin and let PQR be an arbitrary triangle. The point S is such that \[\overrightarrow{OP} \cdot \overrightarrow{OQ} + \overrightarrow{OR} \cdot \overrightarrow{OS} = \overrightarrow{OR} \cdot \overrightarrow{OP} + \overrightarrow{OQ} \cdot \overrightarrow{OS} = \overrightarrow{OQ} \cdot \overrightarrow{OR} + \overrightarrow{OP} \cdot \overrightarrow{OS}\]Then the triangle PQR has S as its

[JEE Main 2019] Let \(\vec{a}\) and \(\vec{b}\) be unit vectors and \(\alpha\) be the angle between them. Then \(\vec{a}+\vec{b}\) is a unit vector if

Let a, b and c be three unit vectors, out of which vectors b and c are non-parallel. If α and β are the angles which vector a makes with vectors b and c respectively and a × (b × c) = \frac{1}{2}b, then |a − b| is equal to

Let $\vec{a} = 2\hat{i}-7\hat{j}+5\hat{k}$, $\vec{b} = \hat{i}+\hat{k}$ and $\vec{c} = \hat{i}+2\hat{j}-3\hat{k}$ be three given vectors. If $\vec{r}$ is a vector such that $\vec{r}\times\vec{a} = \vec{c}\times\vec{a}$ and $\vec{r}\cdot\vec{b}=0$, then $|\vec{r}|$ is equal to:

Let $\vec{a},\vec{b},\vec{c}$ be three vectors such that $\vec{a}\times\vec{b}=2(\vec{a}\times\vec{c})$. If $|\vec{a}|=1$, $|\vec{b}|=4$, $|\vec{c}|=2$, and the angle between $\vec{b}$ and $\vec{c}$ is $60^\circ$, then $|\vec{a}\cdot\vec{c}|$ is equal to:

Let $\vec{a}$, $\vec{b}$ and $\vec{c}$ be three non zero vectors such that $\vec{b}\cdot\vec{c}=0$ and $\vec{a}\times(\vec{b}\times\vec{c}) = \frac{\vec{b}-\vec{c}}{2}$. If $\vec{d}$ be a vector such that $\vec{b}\cdot\vec{d} = \vec{a}\cdot\vec{b}$, then $(\vec{a}\times\vec{b})\cdot(\vec{c}\times\vec{d})$ is equal to

Let b and c be vectors such that |b × c| = 2 and |b| = |c| = 1. If 2b − c = λa, find α + β where λ = √(α − β√3).

Let a b = 3 i + j - k and c ^ ^ ^ \to and \tob . If the vector C\to \to be three vectors such that c\to is coplanar with a is perpendicular to \tob and a \to ⋅ c\to = 5, then |c\to| is equal to

Let a \to = ^i + ^j + k, ^ \to \times b . If \to b = 2 i + 2 j + k and d = a ^ ^ ^ \to⋅\to c is a vector such that a \to \to \to| = 8 c = | c | , | c - 2a 2 \to \to \to \to \to \to 2 and the angle between d and c is \pi 4 , then |10 - 3 b ⋅ c | + | d \times c | is equal to

Let a and b be two unit vectors. If the vectors c = a + 2b and d = 5a − 4a are perpendicular to each other, then the angle between a and b is

Let $\vec{a}=-\hat{i}+2\hat{j}+2\hat{k}$, $\vec{b}=8\hat{i}+7\hat{j}-3\hat{k}$ and $\vec{c}$ be a vector such that $\vec{a}\times\vec{c}=\vec{b}$ and $\vec{c}\cdot(\hat{i}+\hat{j}+\hat{k})=4$. Then $|\vec{a}+\vec{c}|^2$ is equal to:

Between the following two statements: Statement I: Let $\vec{a}=\hat{i}+2\hat{j}-3\hat{k}$ and $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$. Then the vector $\vec{r}$ satisfying $\vec{a}\times\vec{r}=\vec{a}\times\vec{b}$ and $\vec{a}\cdot\vec{r}=0$ is of magnitude $\sqrt{10}$. Statement II: In a triangle $ABC$, $\cos2A+\cos2B+\cos2C\geq-\dfrac{3}{2}$.

We have \(|\vec{a}\times\vec{b} - \vec{a}\times\vec{c}|^2 = |\vec{a}\times(\vec{b}-\vec{c})|^2\). If \(\vec{a}\), \(\vec{b}\), \(\vec{c}\) are unit vectors and the angle between \(\vec{b}\) and \(\vec{c}\) is \(\dfrac{\pi}{3}\), and \(\vec{a}\cdot(\vec{b}-\vec{c})=0\), find \(|\vec{a}\times\vec{b}-\vec{a}\times\vec{c}|^2\).

Let c\to be the projection vector of \tob = \lambda i^ + 4k, ^ \to = i^ + 2 j^ + 2k \lambda > 0, on the vector a ^ \to + c\to| = 7, then the area . If |a of the parallelogram formed by the vectors \tob and c\to is ________ \to \to \to

$\vec{u}_1,\vec{u}_2,\vec{u}_3$ coplanar and $\vec{v}_1,\vec{v}_2,\vec{v}_3$ also coplanar (expressions in $a,b,c$). Then $6(a+b+c)$ is equal to

If the components of \to a = \alpha i + \beta j + \gamma k along and perpendicular to b = 3 i + j - k respectively, are ^ ^ ^ ^ ^ ^ 16 11 ^ ^ ^ (3 i + j - k) and 1 11 ^ ^ ^ (-4 i - 5 j - 17k) , then \alpha + \beta + \gamma is equal to : 2 2 2

If the four points, whose position vectors are $3\hat{i}-4\hat{j}+2\hat{k}$, $\hat{i}+2\hat{j}-\hat{k}$, $-2\hat{i}-\hat{j}+3\hat{k}$ and $5\hat{i}-2\alpha\hat{j}+4\hat{k}$ are coplanar, then $\alpha$ is equal to

Given $\vec{b} - 2\vec{c} = \lambda\vec{a}$ where $|\vec{b}| = 2\vec{c}| = |\lambda\vec{a}|$, $|\vec{b}| + 4|\vec{c}| = 4b$, $\vec{c} = \lambda^2\vec{a}$, and $|\vec{b}.\vec{c}| + |\vec{b}.\vec{c}| = |\vec{b}|^2|\vec{c}|^2$. Find the value of $\lambda^2$.

We have \([\vec{a}\times\vec{b}\;\; \vec{b}\times\vec{c}\;\; \vec{c}\times\vec{a}] = \lambda[\vec{a}\vec{b}\cdot\vec{c}]^2\). Find \(\lambda\).

Let the position vectors of three vertices of a triangle be 4p\to + q\to - 3r\to, -5p\to + q\to + 2r\to and 2 p - q + 2 r . If the \to \to \to position vectors of the orthocenter and the circumcenter of the triangle are and \alphap\to + \betaq\to + \gammar\to p +q +r 4 respectively, then \alpha + 2\beta + 5\gamma is equal to :

Let A, B, C be three points in xy-plane, whose position vector are given by \sqrt3^i + ^j, ^i + \sqrt3^j and a^i + (1 - a)^j respectively with respect to the origin O . If the distance of the point C from the line bisecting the angle between - -\to - -\to the vectors OA and OB is , then the sum of all the possible values of a is : 9 \sqrt2

A weight of 13 kg is supported by two strings of lengths 5 and 12. Given that \(13^2 = 5^2 + 12^2\), the angle \(\angle AOB = \dfrac{\pi}{2}\). Using equilibrium conditions, \(T_1\) and \(T_2\) are:

Let $\vec{a}\times\vec{c}=\vec{a}\times\vec{b}$. If $\vec{a}\cdot\vec{c}=-12$, $\vec{c}\cdot(\hat{i}-2\hat{j}+\hat{k})=5$, then $\vec{c}\cdot(\hat{i}+\hat{j}+\hat{k})$ is equal to _______

Let $\vec{a}=2\hat{i}-\hat{j}+\hat{k}$ and $\vec{b}=\lambda\hat{j}+2\hat{k}$, $\lambda\in\mathbb{Z}$. Let $\vec{c}=\vec{a}\times\vec{b}$ and $\vec{d}$ be a vector of magnitude 2 in $yz$-plane. If $|\vec{c}|=\sqrt{53}$, then the maximum possible value of $(\vec{c}\cdot\vec{d})^2$ is equal to:

$\vec{a}$ parallel to intersection of planes through $\hat{i}+\hat{j},\hat{i}+\hat{k}$ and $\hat{i}-\hat{j},\hat{j}-\hat{k}$. $\theta$ angle between $\vec{a}$ and $\vec{b}=2\hat{i}-2\hat{j}+\hat{k}$, $\vec{a}\cdot\vec{b}=6$. $(\theta,|\vec{a}\times\vec{b}|)=$

$\vec{a}=3\hat{i}+\hat{j}-\hat{k}$, $\vec{c}=2\hat{i}-3\hat{j}+3\hat{k}$. $\vec{a}=\vec{b}\times\vec{c}$, $|\vec{b}|^2=50$. Then $|72-|\vec{b}+\vec{c}|^2|$ is equal to __________.

$\vec{d}\perp\vec{a}=2\hat{i}+7\hat{j}-\hat{k}$ and $\vec{b}=3\hat{i}+5\hat{k}$, $\vec{c}\cdot\vec{d}=12$. Then $(-\hat{i}+\hat{j}-\hat{k})\cdot(\vec{c}\times\vec{d})$ is equal to

Let the arc AC of a circle subtend a right angle at the centre O. If the point B on the arc AC , divides the arc AC - -\to - -\to - -\to length of arc AB such that length of arc BC = 1 5 , and OC = \alphaOA + \betaOB, then \alpha + \sqrt2(\sqrt3 - 1)\beta is equal to

The vectors x\mathbf{i} + (x+1)\mathbf{j} + (x+2)\mathbf{k}, (x+3)\mathbf{i} + (x+4)\mathbf{j} + (x+5)\mathbf{k} and (x+6)\mathbf{i} + (x+7)\mathbf{j} + (x+8)\mathbf{k} are coplanar if x is equal to

Let a \pi 3 \to + 2\tob and 3a . If \lambdaa \to - \lambda\tob are perpendicular to each other, then the number of values of \lambda in [-1, 3] is :

If A, B, C, D and E are five coplanar points, then \(\vec{DA} + \vec{DB} + \vec{DC} + \vec{AE} + \vec{BE} + \vec{CE}\) is equal to

If $\vec{a}$, $\vec{b}$, $\vec{c}$ are three non-zero vectors and $\hat{n}$ is a unit vector perpendicular to $\vec{c}$ such that $\vec{a} = \alpha\vec{b} - \hat{n}$, $(\alpha \neq 0)$ and $\vec{b}\cdot\vec{c} = 12$, then $|\vec{c}\times(\vec{a}\times\vec{b})|$ is equal to:

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

Let $A(2\hat{i}+3\hat{j}+5\hat{k}), B(-\hat{i}+3\hat{j}+2\hat{k})$ and $C(\hat{i}+5\hat{j}+\mu\hat{k})$ are vertices of a triangle and its median through $A$ is equally inclined to the positive directions of the axes. Find the value of $2\lambda - \mu$

[JEE Main 2021] If \(|\vec{a}|=|\vec{b}|=|\vec{a}-\vec{b}|=1\), then \(|\vec{a}+\vec{b}|\) is

$\vec{a}$ and $\vec{b}$ are two non-collinear vectors then the points with position vectors $l_1\vec{a} + m_1\vec{b}, l_2\vec{a} + m_2\vec{b}, l_3\vec{a} + m_3\vec{b}$ are collinear then find the value of $\begin{vmatrix} 1 & 1 & 1 \\ l_1 & l_2 & l_3 \\ m_1 & m_2 & m_3 \end{vmatrix}$.

Let \(\vec{a}\) and \(\vec{b}\) be two vectors such that \(|2\vec{a}+3\vec{b}|=|3\vec{a}+\vec{b}|\). Find the angle between \(\vec{a}\) and \(\vec{b}\).

An arc PQ of a circle subtends a right angle at its centre O. The midpoint of the arc PQ is R. If \(\overrightarrow{OP}=\vec{a}\) and \(\overrightarrow{OQ}=\vec{b}\), find \(\overrightarrow{OR}\).

Let $\hat{a}$ and $\hat{b}$ be two unit vectors such that the angle between them is $\dfrac{\pi}{3}$. If $\lambda\hat{a}+2\hat{b}$ and $3\hat{a}-\lambda\hat{b}$ are perpendicular to each other, then the number of values of $\lambda$ in $[-1,3]$ is:

If \(\vec{x} = 3\hat{i} - 6\hat{j} - \hat{k}\), \(\vec{y} = \hat{i} + 4\hat{j} - 3\hat{k}\) and \(\vec{z} = 3\hat{i} - 4\hat{j} - 12\hat{k}\), then the magnitude of the projection of \(\vec{x} \times \vec{y}\) on \(\vec{z}\) is

The vector \(\hat{i} + x\hat{j} + 3\hat{k}\) is rotated through an angle \(\theta\) and doubled in magnitude, then it becomes \(4\hat{i} + (4x-2)\hat{j} + 2\hat{k}\). The values of \(x\) are

Let a ^ be a unit vector perpendicular to the vectors b = ^ \to i - 2 j + 3k and c = 2 i + 3 j - k, and makes an angle ^ ^ ^ ^ ^ of cos -1 (- 1 3 ) with the vector ^i + ^j + k ^ . If ^ a makes an angle of \pi 3 with the vector ^i + \alpha^j + k ^ , then the value of \alpha is :

If the position vectors of the vertices A, B and C of a \(\triangle ABC\) are, respectively, \(4\hat{i}+7\hat{j}+8\hat{k}\), \(2\hat{i}+3\hat{j}+4\hat{k}\) and \(2\hat{i}+5\hat{j}+7\hat{k}\), then the position vector of the point, where the bisector of \(\angle A\) meets BC is

Let \(\vec{a}, \vec{b}, \vec{c}\) be three vectors of magnitude 2, 3, 5 respectively, satisfying \(|[\vec{a},\, \vec{b},\, \vec{c}]| = 30\). If \((2\vec{a} + \vec{b} + \vec{c}) \cdot ((\vec{a} \times \vec{c}) \times (\vec{a} - \vec{c}) + \vec{b}) = k\), then the value of \(\left\lfloor \dfrac{k}{103} \right\rfloor\) is:

Let $\vec{c}$ be the projection vector of $\vec{b}=\lambda\hat{i}+4\hat{k}$, $\lambda>0$, on the vector $\vec{a}=\hat{i}+2\hat{j}+2\hat{k}$. If $|\vec{a}+\vec{c}|=7$, then the area of the parallelogram formed by the vectors $\vec{b}$ and $\vec{c}$ is ________.

Let the position vectors of three vertices of a triangle be $4\vec{p}+\vec{q}-3\vec{r}$, $-5\vec{p}+\vec{q}+2\vec{r}$ and $2\vec{p}-\vec{q}+2\vec{r}$. If the position vectors of the orthocentre and the circumcentre of the triangle are $\dfrac{\vec{p}+\vec{q}+\vec{r}}{4}$ and $\alpha\vec{p}+\beta\vec{q}+\gamma\vec{r}$ respectively, then $\alpha+2\beta+5\gamma$ is equal to:

If the plane faces of a tetrahedon are represented by the equations $\vec{r} \cdot (\vec{i} + \vec{j}) = 0$, $\vec{r} \cdot (n\vec{k} + m\vec{j}) = 0$, $\vec{r} \cdot (m\vec{k} + \vec{i}) = 0$ and $\vec{r} \cdot (\vec{i} + m\vec{j} + n\vec{k}) = p$, then the volume of the tetrahedon is:

In the figure, \(\overrightarrow{AE}\) is the vector component of \(\vec{q}\) on \(\vec{p}\). From \(\triangle ABE\), we have \(\overrightarrow{AB} + \overrightarrow{BE} = \overrightarrow{AE}\). If \(\vec{q} + \vec{r} = \dfrac{(\vec{p}\cdot\vec{q})}{(\vec{p}\cdot\vec{q})}\vec{p}\), find \(\vec{r}\):

Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=-\hat{i}-8\hat{j}+2\hat{k}$ and $\vec{c}=4\hat{i}+c_2\hat{j}+c_3\hat{k}$ be three vectors. If $\vec{b}\times\vec{a}=\vec{c}\times\vec{a}$, and the angle between the vector $\vec{c}$ and the vector $3\hat{i}+4\hat{j}+\hat{k}$ is $\theta$, then the greatest integer less than or equal to $\tan^2\theta$ is: