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Vectors Questions (152)

Let $\vec{v}_1 = 2(\sin\alpha+\cos\alpha)(\hat{i}+\hat{j})$, $\vec{v}_2 = \sin\beta\hat{i}+\cos\beta\hat{j}$. Given $2(\sin\alpha+\cos\alpha)\sin\beta = 3-\cos\beta$, find $3\tan^2\alpha+4\tan^2\beta$.
A ladder of 3m length leans against a wall. The ladder forms a vertical angle of 30° with the wall. The top slides down at 20 cm/s. The bottom slides away at 20 cm/s at time $t$. The average velocity of a person halfway up the ladder for the first $t$ seconds is
Let $ABCDEF$ be a hexagon with unequal side lengths. Let $G$ and $H$ be centroids of $\triangle ACE$ and $\triangle BDF$ respectively. If $\overrightarrow{AB}-\overrightarrow{BC}+\overrightarrow{CD}-\overrightarrow{DE}+\overrightarrow{EF}-\overrightarrow{FA}=\lambda\overrightarrow{GH}$, then $\lambda$ equals
The scalar triple product $[\vec{a}+\vec{b}-\vec{c}\quad\vec{b}+\vec{c}-\vec{a}\quad\vec{c}+\vec{a}-\vec{b}]$ is equal to
Let $\vec{v}_1 = 2(\sin\alpha+\cos\alpha)(\hat{i}+\hat{j})$, $\vec{v}_2 = \sin\beta\hat{i}+\cos\beta\hat{j}$. Given $2(\sin\alpha+\cos\alpha)\sin\beta = 3-\cos\beta$, find $3\tan^2\alpha+4\tan^2\beta$.
Let $ABCDEF$ be a hexagon with unequal side lengths. Let $G$ and $H$ be centroids of $\triangle ACE$ and $\triangle BDF$ respectively. If $\overrightarrow{AB}-\overrightarrow{BC}+\overrightarrow{CD}-\overrightarrow{DE}+\overrightarrow{EF}-\overrightarrow{FA}=\lambda\overrightarrow{GH}$, then $\lambda$ equals
Which must be true: I) $\vec{a}=2\hat{i}+\hat{j}+\hat{k}$, $\vec{b}$ and $\vec{c}$ nonzero such that $|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|$ and $\vec{b}\cdot\vec{c}=0$, then $|\vec{a}+\lambda\vec{c}|\ge|\vec{a}|$ for all $\lambda\in\mathbb{R}$. II) If $\overrightarrow{PQ},\overrightarrow{QR},\overrightarrow{RS},\overrightarrow{ST},\overrightarrow{TU}$ and $\overrightarrow{UP}$ represent sides of regular hexagon, then $\overrightarrow{PQ}\times(\overrightarrow{RS}+\overrightarrow{ST})\ne\vec{0}$. III) Four points $A,B,C,D$ with position vectors $\vec{a},\vec{b},\vec{c},\vec{d}$ are coplanar, then there exist constants $x,y,z,w$ such that $x\vec{a}+y\vec{b}+z\vec{c}+w\vec{d}=\vec{0}$ with $x+y+z+w=0$ but not all zero.
Let $\vec{a}$, $\vec{b}$, $\vec{c}$ be three non-zero vectors satisfying $\vec{a}=\vec{b}\times\vec{c}+2\vec{b}$, where $|\vec{b}|=|\vec{c}|=2$ and $|\vec{a}|\leq4$. The sum of possible values of $|2\vec{a}+\vec{b}+\vec{c}|$ is
Consider the set of eight vectors $V=\{a\hat{i}+b\hat{j}+c\hat{k}: a,b,c\in\{-1,1\}\}$. The number of ways three non-coplanar vectors can be chosen from $V$ equals
A ladder of 3m length leans against a wall. The ladder forms a vertical angle of 30° with the wall. The top slides down at 20 cm/s. The bottom slides away at 20 cm/s at time $t$. The average velocity of a person halfway up the ladder for the first $t$ seconds is
A, B, C, D are four points with position vectors $\vec{a},\vec{b},\vec{c},\vec{d}$ where $\vec{d}=\alpha\vec{a}+\beta\vec{b}+(1-\alpha-\beta)\vec{c}$. The point D lies on the plane ABC. Which is ALWAYS true?
$\vec{a}=2\hat{i}+3\hat{j}-\hat{k}$, $\vec{b}=\hat{i}+2\hat{j}-5\hat{k}$, $\vec{c}=3\hat{i}+5\hat{j}-11\hat{k}$. Then
Let $\vec{a},\vec{b}$ be two non-collinear unit vectors and $\vec{x}=\vec{a}-(\vec{a}\cdot\vec{b})\vec{b}$ and $\vec{y}=\vec{a}\times\vec{b}$. Statement-I: $|\vec{x}|=|\vec{y}|$. Statement-II: $|\vec{y}|=|\vec{x}|+|\vec{x}\cdot\vec{b}|$.
If $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+7\hat{j}+2\hat{k}$, $\vec{x}\cdot\vec{a}=0$ and $\vec{x}\cdot\vec{c}=0$ for some non-zero vector $\vec{x}$, then value of $\vec{a}\cdot(\vec{b}\times\vec{c})$ is
Let $\vec{p},\vec{q}$ and $\vec{r}$ be three unit vectors satisfying $|\vec{p}-\vec{q}|^2+|\vec{q}-\vec{r}|^2+|\vec{r}-\vec{p}|^2=9$. Then $|2\vec{p}+5\vec{q}+5\vec{r}|$ is equal to
If the volume of a parallelepiped whose coterminous edges are $\vec{a}+\vec{b}$, $\vec{b}+\vec{c}$, $\vec{c}+\vec{a}$ is 24, then volume of parallelepiped with edges $\vec{a},\vec{b},\vec{c}$ is
If $\vec{a}=\hat{i}-\hat{j}+\hat{k}$, $\vec{b}=2\hat{i}+\hat{j}-\hat{k}$ and $\vec{c}=\lambda\hat{i}+\hat{j}-\mu\hat{k}$ are coplanar and the projection of $\vec{c}$ on $2\vec{a}+\vec{b}$ is $\sqrt{6}$, then
$\hat{a}$ and $\hat{b}$ are two unit vectors and $\theta$ is the angle between them. Then $|\hat{a}+\hat{b}|^2-|\hat{a}-\hat{b}|^2=$
The scalar triple product $[\vec{a}+\vec{b}-\vec{c}\quad\vec{b}+\vec{c}-\vec{a}\quad\vec{c}+\vec{a}-\vec{b}]$ is equal to
Let $\vec{v}_1 = 2(\sin\alpha+\cos\alpha)(\hat{i}+\hat{j})$, $\vec{v}_2 = \sin\beta\hat{i}+\cos\beta\hat{j}$. Given $2(\sin\alpha+\cos\alpha)\sin\beta = 3-\cos\beta$, find $3\tan^2\alpha+4\tan^2\beta$.
$[(\vec a\times\vec b)\times(\vec b\times\vec c)]\times(\vec c\times\vec a)\cdot\vec b$ equals (for non-coplanar $\vec a,\vec b,\vec c$)
Let $\vec{a}$, $\vec{b}$ be two vectors perpendicular to each other with $|\vec{a}|=2$, $|\vec{b}|=3$ and $\vec{c}\times\vec{a}=\vec{b}$. The least value of $|\vec{c}-\vec{a}|$ is
A vector $\vec{c}$, directed along the internal bisector of the angle between the 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 :
If $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$, $\vec{b}=2\hat{i}+7\hat{j}+2\hat{k}$, $\vec{x}\cdot\vec{a}=0$ and $\vec{x}\cdot\vec{c}=0$ for some non-zero vector $\vec{x}$, then value of $\vec{a}\cdot(\vec{b}\times\vec{c})$ is
Let $\vec{p},\vec{q}$ and $\vec{r}$ be three unit vectors satisfying $|\vec{p}-\vec{q}|^2+|\vec{q}-\vec{r}|^2+|\vec{r}-\vec{p}|^2=9$. Then $|2\vec{p}+5\vec{q}+5\vec{r}|$ is equal to
The vector sum of $\vec{a}$ and $\vec{b}$ trisects the angle $\theta$ between them. If $|\vec{a}| = a; |\vec{b}| = b; a > b$, then:
Let $\vec{a}$, $\vec{b}$ be two vectors perpendicular to each other with $|\vec{a}|=2$, $|\vec{b}|=3$ and $\vec{c}\times\vec{a}=\vec{b}$. The least value of $|\vec{c}-\vec{a}|$ is
If $\hat{u}$ and $\hat{v}$ are non-collinear unit vectors with $|\hat{u}+\hat{v}+2(\hat{u}\times\hat{v})|=2$, then $\dfrac{|\hat{u}-\hat{v}|}{|\hat{u}\times\hat{v}|}$ equals:
Let $\vec{a}=3\hat i-4\hat j+5\hat k$, $\vec{b}=\hat i+2\hat j-2\hat k$, $\vec{c}=5\hat i-12\hat j-13\hat k$. If $\vec{r}\times\vec{b}=\vec{b}\times(\vec{a}-\vec{c})$ and $\vec{r}\cdot(\vec{a}+\vec{c})=0$, then $|\vec{r}-36\vec{b}|^2$ is
Minimum value of the expression $(a-b)^2 + \left(4\sqrt{1+a^2} + 2\sqrt{2b-b^2}\right)^2$ is
Let $\vec{p},\vec{q}$ and $\vec{r}$ be three unit vectors satisfying $|\vec{p}-\vec{q}|^2+|\vec{q}-\vec{r}|^2+|\vec{r}-\vec{p}|^2=9$. Then $|2\vec{p}+5\vec{q}+5\vec{r}|$ is equal to
Let $\vec{a}$, $\vec{b}$ be two vectors perpendicular to each other with $|\vec{a}|=2$, $|\vec{b}|=3$ and $\vec{c}\times\vec{a}=\vec{b}$. The least value of $|\vec{c}-\vec{a}|$ is
Consider the set of eight vectors $V=\{a\hat{i}+b\hat{j}+c\hat{k}: a,b,c\in\{-1,1\}\}$. The number of ways three non-coplanar vectors can be chosen from $V$ equals
Volume of parallelopiped determined by vectors $\vec{a},\vec{b},\vec{c}$ is 5. Then volume determined by $3(\vec{a}+\vec{b})$, $(\vec{b}+\vec{c})$ and $2(\vec{c}+\vec{a})$ is
A ladder of 3m length leans against a wall. The ladder forms a vertical angle of 30° with the wall. The top slides down at 20 cm/s. The bottom slides away at 20 cm/s at time $t$. The average velocity of a person halfway up the ladder for the first $t$ seconds is
Let $ABCDEF$ be a hexagon with unequal side lengths. Let $G$ and $H$ be centroids of $\triangle ACE$ and $\triangle BDF$ respectively. If $\overrightarrow{AB}-\overrightarrow{BC}+\overrightarrow{CD}-\overrightarrow{DE}+\overrightarrow{EF}-\overrightarrow{FA}=\lambda\overrightarrow{GH}$, then $\lambda$ equals
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:
If $\vec{r} = l(\vec{b} \times \vec{c}) + m(\vec{c} \times \vec{a}) + n(\vec{a} \times \vec{b})$ and $[\vec{b}\vec{c}\vec{c}] = 2$, then $l + m + n$ is equal to:
If vectors $\vec{b} = (\tan\alpha, -1, 2\sqrt{\sin\frac{\alpha}{2}})$ and $\vec{c} = (\tan\alpha, \tan\alpha, -\frac{3}{\sqrt{\sin\alpha/2}})$ are orthogonal and vector $\vec{a} = (1, 3, \sin2\alpha)$ makes an obtuse angle with the z-axis then:
If three coterminous edges of a tetrahedron are $\vec{a}, \vec{b}, \vec{c}$ such that $|\vec{a}| = 2, |\vec{b}| = 3, |\vec{c}| = 4$, angle between $\vec{a}$ and $\vec{b}$ is $\frac{\pi}{3}$, $\vec{b}$ and $\vec{c}$ is $\frac{\pi}{4}$ and $\vec{c}$ and $\vec{a}$ is $\frac{\pi}{6}$. The area of the base is $2$ sq. units, then the height of the tetrahedron is:
If $(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d}) \cdot (\vec{c} \times \vec{b}) = 0$ then which of the following is always true:
If $\vec{\alpha}$ and $\vec{\beta}$ be two perpendicular unit vectors such that $\vec{x} = \vec{\beta} - (\vec{a} \times \vec{x})$, then $|\vec{x}|$ is equal to:
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)
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:
If $ABC$ be a triangle of sides $a, b, c$ with position vectors of $A, B, C$ as $\vec{a}, \vec{b}$ and $\vec{c}$ respectively, then the position vector of its incentre is:
The vector sum of $\vec{a}$ and $\vec{b}$ trisects the angle $\theta$ between them. If $|\vec{a}| = a; |\vec{b}| = b; a > b$, then:
A line passes through the points whose position vectors are $\vec{r} + \vec{j} - 2\vec{k}$ and $\vec{r} - 3\vec{j} + \vec{k}$. The position vector of a point on it at a unit distance from the first point is:
If non-zero vectors $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ satisfy $(\vec{a} \times \vec{b}) \cdot \vec{c} = |\vec{a}||\vec{b}||\vec{c}|$ holds then:
If $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ are any four vectors, then $(\vec{a} \times \vec{b}) \times (\vec{c} \times \vec{d})$ is a vector:
$\vec{a}$ and $\vec{b}$ are two unit vectors inclined at an angle $\alpha(\alpha \in [0, \pi])$ to each other and $|\vec{a} + \vec{b}| < 1$ then $\alpha$ can lie in: