List of top Mathematics Questions on Geometry and Vectors

If the vectors \[ \vec a=4\hat{i}+6\hat{j}+\lambda\hat{k},\qquad \vec b=\hat{i}-2\hat{j}-3\hat{k}, \qquad \vec c=4\lambda\hat{i}+\hat{j}-3\hat{k} \] are coplanar and \(\lambda\in\mathbb{Z}\), then \(\vec a\cdot\vec c=\)

In a triangle \(ABC\), if \[ \overrightarrow{AB}=\hat{i}-2\hat{j}+3\hat{k}, \qquad \overrightarrow{BC}=3\hat{i}+2\hat{j}-2\hat{k}, \] then the triangle is

If \[ \vec a=\hat i+p\hat j-3\hat k,\qquad \vec b=2\hat i-3\hat j+q\hat k,\qquad \vec c=\hat i+2\hat j+2\hat k\;(p0) \] are three vectors such that the magnitude of projection of \(\vec a\) on \(\vec c\) is \(3\) and the magnitude of projection of \(\vec b\) on \(\vec c\) is \(2\), then the magnitude of projection of \(\vec a\) on \(\vec b\) is

If \(\vec a\) is a vector perpendicular to the plane containing the vectors \[ 2\hat i-\hat j+3\hat k \quad\text{and}\quad -\hat i+3\hat j+2\hat k, \] then the magnitude of the projection of the vector \[ 3\hat i+2\hat j-\hat k \] on \(\vec a\) is

The straight line given by the equation \[ \vec r=(4\hat{i}+5\hat{j}+\hat{k})+s(4\hat{i}+6\hat{j}+2\hat{k}) \] is coplanar with the straight line given below. Choose the correct option.

If A(1,2,-3), B(2,3,-1), C(3,1,-2) are vertices of triangle ABC, then area is

Given position vectors of A, B, C, D are coplanar, find \(y-x\) for intersection of AB and CD.

If \( \vec{a} = \hat{i}+\hat{j}+\hat{k} \), \( \vec{b} = \hat{i}-\hat{j}+\hat{k} \) and \( \vec{c} = \hat{i}+\hat{j}-\hat{k} \), then match the following: \[ \begin{array}{|c|c|c|} \hline \text{List-I} & & \text{List-II} \\ \hline A & [\vec{a}\ \vec{b}\ \vec{c}] & I: 4 \\ \hline B & |\vec{a}+\vec{b}+\vec{c}|^2 & II: 11 \\ \hline C & \text{Volume of tetrahedron} & III: \dfrac{2}{3} \\ \hline D & |(\vec{a}\times\vec{b})\times(\vec{a}\times\vec{c})| & IV: 4\sqrt{3} \\ \hline & & V: 12 \\ \hline \end{array} \]

If \(\hat i+2\hat j+\hat k,\ a\hat i+3\hat j+2\hat k,\ -\hat i+4\hat j+\beta\hat k\) are the position vectors of three points \(A,B,C\), then the position vector of a point which divides \(BC\) in the ratio \(a+1:\beta\) is

If a vector \(3\hat i-6\hat j+2\hat k\) makes angles \(\alpha,\beta,\gamma\) with the positive \(x,y,z\)-axes respectively, then \(\cos\alpha+\cos^2\beta+7\cos^3\gamma=\)

Let \( \vec{a} = 3\hat{i} - \hat{j} - \hat{k}, \vec{b} = \hat{i} + \hat{j} - 2\hat{k} \) and \( \vec{c} = 2\hat{i} + 2\hat{j} + \hat{k} \). Let \( \vec{d} \) be a vector such that \( |\vec{d}| = \sqrt{2} \) units. If the vector \( \vec{d} \) is coplanar with \( \vec{a}, \vec{b} \) and perpendicular to \( \vec{c} \), then \( \vec{d} = \)

If \( \vec{a} = \hat{i} - \hat{j} + 3\hat{k} \) and \( \vec{b} = 3\hat{i} - 5\hat{j} + 6\hat{k} \), then the magnitude of the projection of \( 2\vec{a} - \vec{b} \) on \( \vec{a} + \vec{b} \) is:

$\overline{V}=2\overline{i}+\overline{j}-\overline{k}$ and $\overline{W}=\overline{i}+3\overline{k}$. If $\overline{U}$ is a unit vector, then the maximum value of the scalar triple product $[\overline{U}\overline{V}\overline{W}]$ is

$\overline{b}$ and $\overline{c}$ are non-collinear vectors and $\overline{a}$ is a vector such that $(\overline{c}\cdot\overline{c})\overline{a}=\overline{c}$. If $\overline{a}\times(\overline{b}\times\overline{c})+(\overline{a}\cdot\overline{b})\overline{b}=(4-2\beta-sin~\alpha)\overline{b}+(\beta^{2}-1)\overline{c}$, then the values of the scalars $\alpha$ and $\beta$ are

If the median AD of \(\Delta ABC\) is bisected at the point E and BE is produced to meet the side AC at F. Then the vector \( \overline{BF} = \)

Let $\overline{a}=x\overline{i}-2\overline{j}+3\overline{k}$, $\overline{b}=-2\overline{i}+x\overline{j}-\overline{k}$ and $\overline{c}=7\overline{i}-2\overline{j}+x\overline{k}$. If $x=x_0$ is the point of the local maxima of $f(x)=\overline{a}\cdot(\overline{b}\times\overline{c})$, then at $x=x_0$, $\overline{a}\cdot\overline{b}+\overline{b}\cdot\overline{c}+\overline{c}\cdot\overline{a}=$

OABCD is a pentagon in which \(OA\) and \(CB\) are parallel and \(OD\) and \(AB\) are parallel. If \[ \overrightarrow{OA}=\overrightarrow{a}, \qquad \overrightarrow{OD}=\overrightarrow{d}, \] and \[ \frac{OA}{CB}=2, \qquad \frac{OD}{AB}=\frac13, \] then \[ \overrightarrow{AD}+\overrightarrow{OC}+\overrightarrow{DC} \] is equal to:

If \( \overline{a}=\overline{i}+\overline{j}+\overline{k} \), \( \overline{a}.\overline{b}=1 \) and \( \overline{a}\times\overline{b}=\overline{j}-\overline{k} \), then \( \overline{b}= \)

Let \( \overline{a}=4\overline{i}+3\overline{j} \) and \( \overline{b} \) be two vectors in XOY plane, and let \( \overline{a} \) be perpendicular to \( \overline{b} \). Then a vector \( \overline{c} \) in the same plane having projections 1 and 2 respectively on \( \overline{a} \) and \( \overline{b} \) is:

O is the origin, \( \overline{OP} \) and \( \overline{OR} \) are vectors making angles \( 45^{\circ} \) and \( 135^{\circ} \) respectively with the positive direction of x-axis, \( |\overline{OP}|=3 \) and \( |\overline{OR}|=4 \). M is the midpoint of PQ in the rectangle OPQR. If OM meets the diagonal PR at T, then \( \overline{OT}= \)

Let O be the origin, $\vec{OP} = \vec{a}$ and $\vec{OQ} = \vec{b}$. If R is the point on $\vec{OP}$ such that $\vec{OP} = 5\vec{OR}$, and M is the point such that $\vec{OQ} = 5\vec{RM}$, then $\vec{PM}$ is equal to :

Let \( \vec a = 2\hat i + 3\hat j + 3\hat k \) and \( \vec b = 6\hat i + 3\hat j + 3\hat k \). Then the square of the area of the triangle with adjacent sides determined by the vectors \( (2\vec a + 3\vec b) \) and \( (\vec a - \vec b) \) is :

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

If $\vec a,\vec b,\vec c$ are $3$ vectors such that \[ \vec b=2\hat i-\hat j,\qquad \vec c=\hat j+2\hat k, \] \[ |\vec a+\vec b|=3,\qquad |\vec a\times(\vec b\times\vec c)|=3\sqrt2 \] and \[ (\vec a,\vec b\times\vec c)=\frac{\pi}{3}, \] then $\vec a\cdot\vec b=$

Let \[ \vec a=4\hat i-\hat j+\alpha\hat k \] and \[ \vec b=\hat i+\alpha\hat j-4\hat k \] be two vectors. If $\alpha_1,\alpha_2$ ($\alpha_1<\alpha_2$) are two different values of $\alpha$ such that \[ (\vec a,\vec b)=\cos^{-1}\left(-\frac{2}{7}\right), \] then \[ \alpha_1+2\alpha_2= \]