List of top Mathematics Questions on Vector Algebra

Let $\vec{a}=2\hat{i}-\hat{j}-\hat{k}$, $\vec{b}=\hat{i}+3\hat{j}-\hat{k}$ and $\vec{c}=2\hat{i}+\hat{j}+3\hat{k}$. Let $\vec{v}$ be the vector in the plane of $\vec{a}$ and $\vec{b}$, such that the length of its projection on the vector $\vec{c}$ is $\dfrac{1}{\sqrt{14}}$. Then $|\vec{v}|$ is equal to

If \(\overline{AB}=\overline{i}+\overline{j}-2\overline{k}\), \(\overline{CB}=2\overline{i}-\overline{j}+\alpha\overline{k}\) \((\alpha\in Z)\) are two sides of a triangle ABC and the angle between these two sides is \(\frac{\pi}{3}\), then the length of its third side is:

Let \(\overline{OA} = \overline{i} + 2\overline{j} + 2\overline{k}\), \(\overline{OB} = 3\overline{i} + 4\overline{k}\). If \(x\overline{i} + y\overline{j} + z\overline{k}\) is the vector along the bisector of \(\angle AOB\) and of length 2 units, then a possible value of \(x + y + z\) is:

If the shortest distance between the two skew lines \(\overline{r}=\overline{i}+\overline{j}+\overline{k}+t(3\overline{i}+2\overline{j}+\overline{k})\) and \(\overline{r}=\overline{i}-\overline{j}+x\overline{k}+s(\overline{i}+2\overline{j}+3\overline{k})\) is at most \(2\sqrt{6}\), then all values of \(x\) lie in the interval:

If the line \(\overline{r}=\overline{a}+t\overline{b}\) lies on the plane \(\overline{r}\cdot\overline{n}=p\), then \(p=\):

If \[ |\vec a|=3,\qquad |\vec b|=4 \] and the angle between \(\vec a\) and \(\vec b\) is \[ \frac{\pi}{6}, \] then \[ \left|(4\vec a+\vec b)\times(\vec a-3\vec b)\right| = \]

If \(A,B,C\) are the vertices of a triangle \(ABC\), \[ AB=2,\qquad BC=3,\qquad CA=4, \] then \[ \overrightarrow{AB}\cdot\overrightarrow{BC} + \overrightarrow{BC}\cdot\overrightarrow{CA} + \overrightarrow{CA}\cdot\overrightarrow{AB} = \]

If the vector \[ \alpha\hat i+\beta\hat j+\hat k \] is along the bisector of the angle between the vectors \[ 2\hat i-\hat j+2\hat k \] and \[ \hat i+2\hat j+2\hat k, \] then \(2\alpha+6\beta=\)

If \[ \vec{OA}=\hat i-6\hat j+9\hat k,\qquad \vec{OB}=\hat i+3\hat j+5\hat k, \qquad \vec{OC}=2\hat i+\beta\hat j+7\hat k \] are the position vectors of three collinear points \(A,B,C\), then the ratio in which \(B\) divides \(AC\) is

The direction cosines of the vector \[ \vec a=a_1\hat i+a_2\hat j+a_3\hat k \] are

Let \[ \vec a=\vec i+2\vec j+\vec k \] and \[ \vec b=2\vec i-\vec j+\vec k \] be two vectors. If vector \[ \vec r=x\vec i+y\vec j+2\vec k \] is along the bisector of angle between \(\vec a\) and \(\vec b\), then \[ |\vec r|= \]

If the points with position vectors \[ x\vec i+2\vec j+y\vec k \] \[ \vec i-2\vec j+2x\vec k \] and \[ 2\vec i+3\vec j-\vec k \] are collinear, then \[ 10x-25y= \]

Let \[ \vec a=2\vec i-\vec j-3\vec k,\qquad \vec b=\vec i+3\vec j-2\vec k,\qquad \vec c=3\vec i-2\vec j+\vec k \] If magnitude of projection of \[ \vec a+\lambda\vec b \] on \(\vec c\) is \[ \frac{10}{\sqrt{14}} \] then sum of squares of magnitudes of all such vectors is

Let \[ \vec a=3\vec i-2\vec j+5\vec k,\qquad \vec b=\vec i+3\vec j-2\vec k \] If \(\vec c\) is a vector such that \[ \vec b\times\vec c=\vec a \] and \[ \vec b\cdot\vec c=5 \] then \[ 14\vec c\times\vec a= \]

Let \[ \vec a=\vec i-2\vec j+2\vec k \] and \[ \vec b=2\vec i+3\vec j-6\vec k \] If \[ \alpha\vec i+\beta\vec j+\gamma\vec k \] is perpendicular to plane of \[ 2\vec a+\vec b \] and \[ \vec b-\vec a \] such that \[ \alpha+\beta+\gamma=46 \] then \[ \alpha-2\beta+3\gamma= \]

Let \(\vec a,\vec b\) be two non-collinear vectors. If \[ \vec r=(x+2y-3)\vec a+(2x-y+1)\vec b \] and \[ \vec R=(3x-y-2)\vec a+(x+3y+2)\vec b \] are vectors such that \[ 2\vec r=m\vec R, \] then \(x+5y=\)

The conjugate of $(1+i)^3$ is:

If $z_1 = 4i^{40} - 5i^{35} + 6i^{17} + 2$ and $z_2 = -1 + i$, then $|z_1 + z_2| =$

If $\frac{x}{(x-1)^{2}(x+2)}=\frac{A}{(x-1)^{2}}+\frac{B}{9(x-1)}+\frac{C}{(x+2)}$ then $A+B=$

If $\frac{x+4}{(x+2)^{2}(x+3)}=\frac{A}{(x+2)^{2}}+\frac{B}{(x+2)}+\frac{C}{(x+3)}$ then $A+B+C=$

Let \(\hat{u}\) and \(\hat{v}\) be unit vectors inclined at an acute angle such that \(|\hat{u} \times \hat{v}| = \frac{\sqrt{3}}{2}\). If \(\vec{A} = \lambda \hat{u} + \hat{v} + (\hat{u} \times \hat{v})\), then \(\lambda\) is equal to:

Two adjacent sides of a parallelogram PQRS are given by \( \overrightarrow{PQ} = \hat{i} + \hat{j} + \hat{k} \) and \( \overrightarrow{PS} = \hat{i} - \hat{j} \). If the side PS is rotated about the point P by an acute angle \( \alpha \) in the plane of the parallelogram so that it becomes perpendicular to the side PQ, then \( \sin^2 \left( \frac{5\alpha}{2} \right) - \sin^2 \left( \frac{\alpha}{2} \right) \) is equal to:

Let the vectors \( \mathbf{a} = -\hat{i} + \hat{j} + 3\hat{k} \) and \( \mathbf{b} = \hat{i} + 3\hat{j} + \hat{k} \). For some \( \lambda, \mu \in \mathbb{R} \), let \( \mathbf{c} = \lambda \mathbf{a} + \mu \mathbf{b} \). If \( \mathbf{c} \cdot (3\hat{i} - 6\hat{j} + 2\hat{k}) = 10 \) and \( \mathbf{c} \cdot (\hat{i} + \hat{j} + \hat{k}) = -2 \), then \( |\mathbf{c}|^2 \) is equal to:

Let \( PQR \) be a triangle such that \[ \vec{PQ}=-2\hat i-\hat j+2\hat k,\quad \vec{PR}=a\hat i+b\hat j-4\hat k,\ a,b\in\mathbb{Z}. \] Let \( S \) be the point on \( QR \) which is equidistant from the lines \( PQ \) and \( PR \). If \[ |\vec{PR}|=9 \quad \text{and} \quad \vec{PS}=\hat i-7\hat j+2\hat k, \] then the value of \( 3a-4b \) is:

For three unit vectors \( \vec a, \vec b, \vec c \) satisfying \[ |\vec a-\vec b|^2 + |\vec b-\vec c|^2 + |\vec c-\vec a|^2 = 9 \] and \[ |2\vec a + k\vec b + k\vec c| = 3, \] the positive value of \( k \) is: