List of top Mathematics Questions on Geometry and Vectors

Prove by vector method an angle in a semi-circle is a right angle.

The vector equation of the XY-plane is

If the angle between the planes ax-y+3z=2a and 3x+ay+z=3a is \( \frac{\pi}{3} \) then the direction ratios of the line perpendicular to the plane (a+2)x+(a-4)y+2az=a are

If \(\alpha\) is the angle between any two diagonals of a cube and \(\beta\) is the angle between a diagonal of a cube and a diagonal of its face, which intersects this diagonal of the cube then \( \cos\alpha + \cos^2\beta = \)

The equation of the locus of a point whose distance from XY-plane is twice its distance from Z-axis is

If \( \vec{a} = \vec{i}-2\vec{j}+2\vec{k} \), \( \vec{b} = 6\vec{i}+3\vec{j}-2\vec{k} \), \( \vec{c} = -4\vec{i}+3\vec{j}+12\vec{k} \) are three vectors then the value of the expression is...

If the volume of a tetrahedron having \( \vec{i}+2\vec{j}-3\vec{k} \), \( 2\vec{i}+\vec{j}-3\vec{k} \) and \( 3\vec{i}-\vec{j}+p\vec{k} \) as its coterminous edges is 2, then the values of p are the roots of the equation

If \( \vec{a} \) and \( \vec{b} \) are two vectors such that \( |\vec{a}|=|\vec{b}|=\sqrt{6} \) and \( \vec{a} \cdot \vec{b} = -1 \), then \( |\vec{a} \times \vec{b}| \sin(\vec{a}, \vec{b}) = \)

Let \( \vec{a} \) and \( \vec{b} \) be two vectors such that \( |\vec{a}| = |\vec{b}| \) and \( |\vec{a}+2\vec{b}| = |2\vec{a}-\vec{b}| \). If \( \vec{c} \) is a vector parallel to \( \vec{a} \) then the angle between \( \vec{b} \) and \( \vec{c} \) is

ABCD is a tetrahedron. \( \vec{i}-2\vec{j}+3\vec{k} \), \( -2\vec{i}+\vec{j}+3\vec{k} \), \( 3\vec{i}+2\vec{j}-\vec{k} \) are the position vectors of the points A, B, C respectively. \( -\vec{i}+2\vec{j}-3\vec{k} \) is the position vector of the centroid of the triangular face BCD. If G is the centroid of the tetrahedron, then GD =

If the equation of the plane passing through the points (2,1,2), (1,2,1) and perpendicular to the plane \( 2x-y+2z=1 \) is \( ax+by+cz+d=0 \) then \( \frac{a+b}{c+d} = \)

The shortest distance between the lines \( \bar{r} = (3\bar{i} - 5\bar{j} + 2\bar{k}) + t(4\bar{i} + 3\bar{j} - \bar{k}) \) and \( \bar{r} = (\bar{i} + 2\bar{j} - 4\bar{k}) + s(6\bar{i} + 3\bar{j} - 2\bar{k}) \) is

If \( \bar{a} = \bar{i} - 2\bar{j} - 2\bar{k} \) and \( \bar{b} = 2\bar{i} + \bar{j} + 2\bar{k} \) are two vectors then \( (\bar{a} + 2\bar{b}) \times (3\bar{a} - \bar{b}) = \)

If the direction cosines of two lines satisfy the equations \( 2l+m-n=0 \), \( l^2-2m^2+n^2=0 \) and \( \theta \) is the angle between the lines then \( \cos\theta = \)

If A(0,3,4), B(1,5,6), C(-2,0,-2) are the vertices of a triangle ABC and the bisector of angle A meets the side BC at D, then AD =

If \( \bar{a} \) and \( \bar{b} \) are two vectors such that \( |\bar{a}|=5 \), \( |\bar{b}|=12 \) and \( |\bar{a}-\bar{b}|=13 \) then \( |2\bar{a}+\bar{b}| = \)

The point of intersection of the line joining the points \( \bar{i} + 2\bar{j} + \bar{k} \), \( 2\bar{i} - \bar{j} - \bar{k} \) and the plane passing through the points \( \bar{i}, 2\bar{j}, 3\bar{k} \) is

The position vectors of two points A and B are \( \bar{i} + 2\bar{j} + 3\bar{k} \) and \( 7\bar{i} - \bar{k} \) respectively. The point P with position vector \( -2\bar{i} + 3\bar{j} + 5\bar{k} \) is on the line AB. If the point Q is the harmonic conjugate of P, then the sum of the scalar components of the position vector of Q is

\( \bar{a}, \bar{b}, \bar{c} \) are three unit vectors such that \( x\bar{a} + y\bar{b} + z\bar{c} = p(\bar{b} \times \bar{c}) + q(\bar{c} \times \bar{a}) + r(\bar{a} \times \bar{b}) \). If \( (\bar{a},\bar{b})=(\bar{b},\bar{c})=(\bar{c},\bar{a})=\frac{\pi}{3} \), \( (\bar{a}, \bar{b} \times \bar{c})=\frac{\pi}{6} \) and \( \bar{a}, \bar{b}, \bar{c} \) form a right-handed system, then \( \frac{x+y+z}{p+q+r} = \)

Let A be a point having position vector \( \vec{i}-3\vec{j} \) and \( \bar{r} = (\vec{i}-3\vec{j}) + t(\vec{j}-2\vec{k}) \) be a line. If P is a point on this line and is at a minimum distance from the plane \( \bar{r} \cdot (2\vec{i}+3\vec{j}+5\vec{k}) = 0 \), then the equation of the plane through P and perpendicular to AP is

If \( a=|\bar{a}| \); \( b=|\bar{b}| \) then \( \left(\frac{\bar{a}}{a^2} - \frac{\bar{b}}{b^2}\right)^2 = \)

The four points whose position vectors are given by \( 2\bar{a}+3\bar{b}-\bar{c} \), \( \bar{a}-2\bar{b}+3\bar{c} \), \( 3\bar{a}+4\bar{b}-2\bar{c} \) and \( \bar{a}-6\bar{b}+6\bar{c} \) are

A, B, C, D are any four points. If E and F are mid points of AC and BD respectively, then \( \vec{AB}+\vec{CB}+\vec{CD}+\vec{AD} = \)